Previous Talks
Florian Karl Richter, Sums and products in sets of positive density
(EPFL)
Florian Karl Richter, Sums and products in sets of positive density
(EPFL)
Thursday, March 13, 2025 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 14, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 13, 2025 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 14, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Hindman's conjecture states that for any finite coloring of the integers, there exist natural numbers x and y such that x, y, x+y, xy all have the same color. This conjecture remains open, with its difficulty stemming from the challenge of controlling arithmetic structures that simultaneously involve both addition and multiplication. In this talk, we will discuss how arithmetic configurations as the ones appearing in Hindman’s conjecture are governed by the local Host-Kra uniformity norms. Our approach relies on tools and ideas from multiplicative number theory. This allows us to establish a density analogue of a special case of a theorem of Moreira and to resolve a conjecture of Moreira.
Abstract: Hindman's conjecture states that for any finite coloring of the integers, there exist natural numbers x and y such that x, y, x+y, xy all have the same color. This conjecture remains open, with its difficulty stemming from the challenge of controlling arithmetic structures that simultaneously involve both addition and multiplication. In this talk, we will discuss how arithmetic configurations as the ones appearing in Hindman’s conjecture are governed by the local Host-Kra uniformity norms. Our approach relies on tools and ideas from multiplicative number theory. This allows us to establish a density analogue of a special case of a theorem of Moreira and to resolve a conjecture of Moreira.
Péter Varjú, Lifting in special linear groups
(University of Cambridge)
Péter Varjú, Lifting in special linear groups
(University of Cambridge)
Thursday, March 6, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 7, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 6, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 7, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Given an element in SL_n(Z/qZ), what is the smallest element of SL_n(Z) that projects to it? We show that a lift with entries bounded by O(q^2 log q) always exists, and that the exponent 2 is best possible. Time permitting we may discuss the analogous problem of finding integer matrices with prescribed determinant that approximates a given matrix with real entries. Joint work with Amitay Kamber.
Abstract: Given an element in SL_n(Z/qZ), what is the smallest element of SL_n(Z) that projects to it? We show that a lift with entries bounded by O(q^2 log q) always exists, and that the exponent 2 is best possible. Time permitting we may discuss the analogous problem of finding integer matrices with prescribed determinant that approximates a given matrix with real entries. Joint work with Amitay Kamber.
Link to recording (YouTube)
Link to slides
Shaoshi Chen, Rational-Transcendental Dichotomy Theorems on Power Series with Arithmetic Restrictions
(Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Shaoshi Chen, Rational-Transcendental Dichotomy Theorems on Power Series with Arithmetic Restrictions
(Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Thursday, February 27, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 28, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 27, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 28, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In 1906, Fatou proved a rational-transcendental dichotomy theorem on power series with integer coefficients. This theorem has been generalized to a broader class of power series whose coefficients satisfy certain arithmetic restrictions. This talk will first recall some rational-transcendental dichotomy theorems in number fields and then present some more recent theorems in the context of power series rings and differential fields, along with related conjectures and open problems. This talk is based on joint work with Jason Bell, Ehsaan Hossain, Khoa Nguyen, and Umberto Zannier.
Abstract: In 1906, Fatou proved a rational-transcendental dichotomy theorem on power series with integer coefficients. This theorem has been generalized to a broader class of power series whose coefficients satisfy certain arithmetic restrictions. This talk will first recall some rational-transcendental dichotomy theorems in number fields and then present some more recent theorems in the context of power series rings and differential fields, along with related conjectures and open problems. This talk is based on joint work with Jason Bell, Ehsaan Hossain, Khoa Nguyen, and Umberto Zannier.
Link to recording (YouTube)
Link to slides
Valérie Berthé, A symbolic approach to bounded remainder sets
(IRIF, Université Paris Cité)
Valérie Berthé, A symbolic approach to bounded remainder sets
(IRIF, Université Paris Cité)
Thursday, February 20, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 21, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 20, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 21, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A bounded remainder set is a set with bounded (local) discrepancy. We discuss dynamical and symbolic approaches to the study of bounded remainder sets for Kronecker sequences. We focus on the case of Pisot parameters and show how to construct bounded remainder sets in terms of multidimensional continued fractions. We also discuss convergence issues for multidimensional continued fractions in terms of their Lyapounov exponents.
Abstract: A bounded remainder set is a set with bounded (local) discrepancy. We discuss dynamical and symbolic approaches to the study of bounded remainder sets for Kronecker sequences. We focus on the case of Pisot parameters and show how to construct bounded remainder sets in terms of multidimensional continued fractions. We also discuss convergence issues for multidimensional continued fractions in terms of their Lyapounov exponents.
This is joint work with W. Steiner and J. Thuswaldner.
This is joint work with W. Steiner and J. Thuswaldner.
Link to recording (YouTube)
Link to slides
Emmanuel Ullmo, Bi-$\overline{Q}$-Structures on Hermitian Symmetric Spaces and quadratic relations between CM periods
(IHES)
Emmanuel Ullmo, Bi-$\overline{Q}$-Structures on Hermitian Symmetric Spaces and quadratic relations between CM periods
(IHES)
Thursday, February 13, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 14, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 13, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 14, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We define a natural bi-$\overline{\Q}$-structure on the tangent space at a CM point on a Hermitian locally symmetric domain. We prove that this bi-$\overline{\Q}$-structure decomposes into the direct sum of 1-dimensional bi-$\overline{\Q}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $A_g$. We propose an "Hyperbolic Analytic Subspace" Conjecture, which is the analogue of Wüstholz’s Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $A_g$ , implies that all quadratic $\overline{\Q}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones. We then show that the elementary quadratic relations between CM periods are at the heart of the theory: For any CM abelian variety $A$, there exists an abelian variety $B$ such that all the algebraic relations among CM periods on $A\times B$, induced by Hodge cycles, are generated by these elementary quadratic relations.
Abstract: We define a natural bi-$\overline{\Q}$-structure on the tangent space at a CM point on a Hermitian locally symmetric domain. We prove that this bi-$\overline{\Q}$-structure decomposes into the direct sum of 1-dimensional bi-$\overline{\Q}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $A_g$. We propose an "Hyperbolic Analytic Subspace" Conjecture, which is the analogue of Wüstholz’s Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $A_g$ , implies that all quadratic $\overline{\Q}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones. We then show that the elementary quadratic relations between CM periods are at the heart of the theory: For any CM abelian variety $A$, there exists an abelian variety $B$ such that all the algebraic relations among CM periods on $A\times B$, induced by Hodge cycles, are generated by these elementary quadratic relations.
Joint work with Ziyang Gao and Andrei Yafaev.
Link to recording (YouTube)
Link to slides
Alex Kontorovich, Spanning Trees of Graphs
(Rutgers University)
Alex Kontorovich, Spanning Trees of Graphs
(Rutgers University)
Thursday, February 6, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 7, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 6, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 7, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We prove the exponential growth of the cardinality of the set of numbers of spanning trees in graphs, answering a question of Sedlacek from 1969. The proof uses a connection with continued fractions, Diophantine approximation, and advances towards Zaremba’s conjecture. This is joint work with Swee Hong Chan and Igor Pak.
Abstract: We prove the exponential growth of the cardinality of the set of numbers of spanning trees in graphs, answering a question of Sedlacek from 1969. The proof uses a connection with continued fractions, Diophantine approximation, and advances towards Zaremba’s conjecture. This is joint work with Swee Hong Chan and Igor Pak.
Link to recording (YouTube)
Link to slides
Hector Pasten, Effective Mordell for curves with enough automorphisms
(Pontificia Universidad Católica de Chile)
Hector Pasten, Effective Mordell for curves with enough automorphisms
(Pontificia Universidad Católica de Chile)
Thursday, January 30, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 31, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 30, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 31, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The effective Mordell conjecture asks for an algorithm to compute the rational points of curves of genus g>1 defined over number fields. At present this is open. While there are methods derived from Chabauty--Coleman--Kim that in practice work extremely well under some assumptions, these methods are not known to terminate. Our main result is an explicit and computable height bound for rational points of curves with "enough automorphisms", which gives a practical algorithm that terminates when the relevant hypothesis is satisfied; we will present an example. Our methods build on Arakelov geometry and sphere packing. This is joint work with Natalia Garcia-Fritz.
Abstract: The effective Mordell conjecture asks for an algorithm to compute the rational points of curves of genus g>1 defined over number fields. At present this is open. While there are methods derived from Chabauty--Coleman--Kim that in practice work extremely well under some assumptions, these methods are not known to terminate. Our main result is an explicit and computable height bound for rational points of curves with "enough automorphisms", which gives a practical algorithm that terminates when the relevant hypothesis is satisfied; we will present an example. Our methods build on Arakelov geometry and sphere packing. This is joint work with Natalia Garcia-Fritz.
Link to recording (YouTube)
Link to slides
Katherine E. Stange, The local-global conjecture for Apollonian circle packings is false
(University of Colorado, Boulder)
Katherine E. Stange, The local-global conjecture for Apollonian circle packings is false
(University of Colorado, Boulder)
Thursday, January 23, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 24, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 23, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 24, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group', a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.
Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group', a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.
Link to recording (YouTube)
Julia Stadlmann, Primes in arithmetic progressions to smooth moduli
(Stanford University)
Julia Stadlmann, Primes in arithmetic progressions to smooth moduli
(Stanford University)
Thursday, January 16, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 17, 2025 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 16, 2025 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 17, 2025 (12am CST, 3am AEDT, 5am NZDT)
Abstract: For large x and coprime a and q, the arithmetic progression n = a mod q contains approximately pi(x)/phi(q) primes up to x. For which moduli q can we prove that this approximation has small error terms? In this talk, I will focus on results for smooth moduli, which were a key ingredient in Zhang's proof of bounded gaps between primes and later improvements of Polymath. Following arguments of the Polymath project, I will sketch how better equidistribution estimates for primes in APs are linked to stronger bounds on the infimum limit of gaps between m consecutive primes. I will then show how a refinement of the q-van der Corput method can be used to improve on Polymath's equidistribution estimates and thus to obtain better bounds on short gaps between 3 or more primes.
Abstract: For large x and coprime a and q, the arithmetic progression n = a mod q contains approximately pi(x)/phi(q) primes up to x. For which moduli q can we prove that this approximation has small error terms? In this talk, I will focus on results for smooth moduli, which were a key ingredient in Zhang's proof of bounded gaps between primes and later improvements of Polymath. Following arguments of the Polymath project, I will sketch how better equidistribution estimates for primes in APs are linked to stronger bounds on the infimum limit of gaps between m consecutive primes. I will then show how a refinement of the q-van der Corput method can be used to improve on Polymath's equidistribution estimates and thus to obtain better bounds on short gaps between 3 or more primes.
Link to recording (YouTube)
Link to slides
David Zureick-Brown, ell-adic Images of Galois for Elliptic Curves over Q
(Amherst College)
David Zureick-Brown, ell-adic Images of Galois for Elliptic Curves over Q
(Amherst College)
Thursday, December 19, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 20, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 19, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 20, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated to an elliptic curve (equivalently, classification of all rational points on certain modular curves XH). The main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.
Abstract: I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated to an elliptic curve (equivalently, classification of all rational points on certain modular curves XH). The main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.
I will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.
I will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.
Link to recording (YouTube)
Link to slides
Katharine Woo, Manin's conjecture for Châtelet surfaces
(Princeton University)
Katharine Woo, Manin's conjecture for Châtelet surfaces
(Princeton University)
Thursday, December 12, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 13, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 12, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 13, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.
Abstract: We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.
Link to recording (YouTube)
Link to slides
Asif Zaman, The least prime in the Chebotarev density theorem for symmetric groups and more
(University of Toronto)
Asif Zaman, The least prime in the Chebotarev density theorem for symmetric groups and more
(University of Toronto)
Thursday, December 5, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, December 6, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 5, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, December 6, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Let $K/k$ be a Galois extension of number fields with Galois group $G$. For a conjugacy class $C$ of $G$, the least unramified prime with Frobenius element in $C$ is known to be at most a fixed absolute power of the discriminant of $K$ due to the celebrated work of Lagarias, Montgomery, and Odlyzko (1979). This theorem has been extensively studied with the primary method exploiting statistics of zeros of L-functions. The current record for the exponent is 16 due to Kadiri, Ng, and Wong (2019). For $G = S_n$, I will describe a method based on detecting sign changes that improves this exponent to decay exponentially with $n$ as $n \to \infty$. The ideas also apply to other groups $G$ and conjugacy invariant subsets $C$.
Abstract: Let $K/k$ be a Galois extension of number fields with Galois group $G$. For a conjugacy class $C$ of $G$, the least unramified prime with Frobenius element in $C$ is known to be at most a fixed absolute power of the discriminant of $K$ due to the celebrated work of Lagarias, Montgomery, and Odlyzko (1979). This theorem has been extensively studied with the primary method exploiting statistics of zeros of L-functions. The current record for the exponent is 16 due to Kadiri, Ng, and Wong (2019). For $G = S_n$, I will describe a method based on detecting sign changes that improves this exponent to decay exponentially with $n$ as $n \to \infty$. The ideas also apply to other groups $G$ and conjugacy invariant subsets $C$.
This talk is based on joint work with Peter Cho and Robert Lemke Oliver.
This talk is based on joint work with Peter Cho and Robert Lemke Oliver.
Link to slides
Julian Demeio, The Grunwald Problem for solvable groups
(University of Bath)
Julian Demeio, The Grunwald Problem for solvable groups
(University of Bath)
Thursday, November 28, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 29, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 28, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 29, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups.
Abstract: Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups.
Link to recording (YouTube)
Link to slides
Martin Orr, Height bounds for very unlikely intersections in abelian varieties using G-functions
(University of Manchester)
Martin Orr, Height bounds for very unlikely intersections in abelian varieties using G-functions
(University of Manchester)
Thursday, November 21, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 22, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 21, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 22, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A special case of the Zilber-Pink conjecture, proved by Habegger and Pila, states that a generic curve C in an abelian variety A has only finitely many "unlikely intersections", that is, intersections of C with subgroups of A of codimension at least 2. One important ingredient in the proof is a bound for the height of these intersection points. In this talk, I will discuss a new method of proving such a height bound for intersections with subgroups of large codimension ("very unlikely intersections"), using ideas of Bombieri and André about G-functions. A benefit of this method is that it is in principle effective.
Abstract: A special case of the Zilber-Pink conjecture, proved by Habegger and Pila, states that a generic curve C in an abelian variety A has only finitely many "unlikely intersections", that is, intersections of C with subgroups of A of codimension at least 2. One important ingredient in the proof is a bound for the height of these intersection points. In this talk, I will discuss a new method of proving such a height bound for intersections with subgroups of large codimension ("very unlikely intersections"), using ideas of Bombieri and André about G-functions. A benefit of this method is that it is in principle effective.
Link to recording (YouTube)
Anders Södergren, Low-lying zeros in families of modular form L-functions
(Chalmers University of Technology)
Anders Södergren, Low-lying zeros in families of modular form L-functions
(Chalmers University of Technology)
Thursday, November 14, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 15, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 14, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 15, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In this talk, I will discuss the distribution of zeros in families of L-functions. The focus will be on ideas and results related to the Katz-Sarnak heuristic for the statistics of low-lying zeros, that is, zeros that are located close to the real axis. In particular, I will report on joint work with Martin Čech, Lucile Devin, Daniel Fiorilli and Kaisa Matomäki on extended density theorems in certain families of L-functions attached to holomorphic modular forms or Maass forms.
Abstract: In this talk, I will discuss the distribution of zeros in families of L-functions. The focus will be on ideas and results related to the Katz-Sarnak heuristic for the statistics of low-lying zeros, that is, zeros that are located close to the real axis. In particular, I will report on joint work with Martin Čech, Lucile Devin, Daniel Fiorilli and Kaisa Matomäki on extended density theorems in certain families of L-functions attached to holomorphic modular forms or Maass forms.
Link to recording (YouTube)
Dubi Kelmer, Values of quadratic forms at integer points
(Boston College)
Dubi Kelmer, Values of quadratic forms at integer points
(Boston College)
Thursday, November 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 8, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time)
Friday, November 8, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The Oppenheim conjecture, proved by Margulis, states that the values at integers of an indefinite irrational quadratic form in 3 or more variables are dense on the real line. In this talk I will survey some recent results regarding effectiveness of this result for homogenous as well as inhomogeneous forms.
Abstract: The Oppenheim conjecture, proved by Margulis, states that the values at integers of an indefinite irrational quadratic form in 3 or more variables are dense on the real line. In this talk I will survey some recent results regarding effectiveness of this result for homogenous as well as inhomogeneous forms.
Link to recording (YouTube)
Link to slides
Manfred Einsiedler, Effective Equidistribution of semisimple adelic periods
(ETH Zürich)
Manfred Einsiedler, Effective Equidistribution of semisimple adelic periods
(ETH Zürich)
Thursday, October 31, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 1, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, October 31, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 1, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We will discuss an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients, obtained in ongoing joint work with E. Lindenstrauss, A. Mohammadi, and A. Wieser. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. The proof uses dynamical arguments, Clozel's property (tau), Prasad's volume formula, an effective closing lemma, and a novel effective generation result for subgroups. The latter in turn relies on an effective version of Greenberg's theorem.
Abstract: We will discuss an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients, obtained in ongoing joint work with E. Lindenstrauss, A. Mohammadi, and A. Wieser. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. The proof uses dynamical arguments, Clozel's property (tau), Prasad's volume formula, an effective closing lemma, and a novel effective generation result for subgroups. The latter in turn relies on an effective version of Greenberg's theorem.
We apply the above to the problem of establishing a local-global principle for representations of quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh.
We apply the above to the problem of establishing a local-global principle for representations of quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh.
Link to recording (YouTube)
Link to slides
Niclas Technau, Smooth discrepancy and Littlewood’s conjecture
(University of Bonn)
Niclas Technau, Smooth discrepancy and Littlewood’s conjecture
(University of Bonn)
Thursday, October 24, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 24, 2024 (2am AEDT, 4am NZDT)
Thursday, October 24, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 24, 2024 (2am AEDT, 4am NZDT)
Abstract: Let \boldsymbol \alpha \in [0,1]^d. This talk concerns fine-scale statistics of the Kronecker sequences (n \boldsymbol \alpha \: \mathrm{mod} \: 1)_{n=1}^\infty. Reporting on joint work with Sam Chow, I will discuss a local-to-global principle. The principle relates the smooth discrepancy (a global, analytic quantity) of Kronecker sequences to their multiplicative diophantine approximability (a local, arithmetic property). This opens up a new avenue of attack for a conjecture of Littlewood.
Abstract: Let \boldsymbol \alpha \in [0,1]^d. This talk concerns fine-scale statistics of the Kronecker sequences (n \boldsymbol \alpha \: \mathrm{mod} \: 1)_{n=1}^\infty. Reporting on joint work with Sam Chow, I will discuss a local-to-global principle. The principle relates the smooth discrepancy (a global, analytic quantity) of Kronecker sequences to their multiplicative diophantine approximability (a local, arithmetic property). This opens up a new avenue of attack for a conjecture of Littlewood.
Link to recording (YouTube)
Link to slides
Kevin Ford, On the theory of prime producing sieves, part 2
(University of Illinois at Urbana-Champaign)
Kevin Ford, On the theory of prime producing sieves, part 2
(University of Illinois at Urbana-Champaign)
Thursday, October 17, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 18, 2024 (2am AEDT, 4am NZDT)
Thursday, October 17, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 18, 2024 (2am AEDT, 4am NZDT)
Abstract: The closest thing to a general method for counting primes in a set is the method of Type I/II sums. This allows one to obtain an asymptotic formula (or perhaps a non-trivial lower bound) for the number of primes in the set, provided one has sufficiently good estimates for certain auxiliary sums.
Abstract: The closest thing to a general method for counting primes in a set is the method of Type I/II sums. This allows one to obtain an asymptotic formula (or perhaps a non-trivial lower bound) for the number of primes in the set, provided one has sufficiently good estimates for certain auxiliary sums.
Unfortunately what counts as 'sufficiently good' is poorly understood, as are the limits of this approach. In this talk I'll talk a new framework (joint with James Maynard) which allows us to prove various necessary and sufficient conditions, focusing on methods for constructing sets that satisfy the Type I and Type II bounds yet contain no primes. In particular, I will go into some detail about how to prove that a substantial 'Type II range' is necessary to deduce the existence of primes in a set.
Unfortunately what counts as 'sufficiently good' is poorly understood, as are the limits of this approach. In this talk I'll talk a new framework (joint with James Maynard) which allows us to prove various necessary and sufficient conditions, focusing on methods for constructing sets that satisfy the Type I and Type II bounds yet contain no primes. In particular, I will go into some detail about how to prove that a substantial 'Type II range' is necessary to deduce the existence of primes in a set.
Link to recording (YouTube)
Link to slides
James Maynard, On the theory of prime producing sieves, part 1
(University of Oxford)
James Maynard, On the theory of prime producing sieves, part 1
(University of Oxford)
Thursday, October 10, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 11, 2024 (2am AEDT, 4am NZDT)
Thursday, October 10, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 11, 2024 (2am AEDT, 4am NZDT)
Abstract: The closest thing to a general method for counting primes in a set is the method of Type I/II sums. This allows one to obtain an asymptotic formula (or perhaps a non-trivial lower bound) for the number of primes in the set, provided one has sufficiently good estimates for certain auxiliary sums.
Abstract: The closest thing to a general method for counting primes in a set is the method of Type I/II sums. This allows one to obtain an asymptotic formula (or perhaps a non-trivial lower bound) for the number of primes in the set, provided one has sufficiently good estimates for certain auxiliary sums.
Unfortunately what counts as 'sufficiently good' is poorly understood, as are the limits of this approach. In this talk, I'll discuss a new framework (joint with Kevin Ford) which allows us to prove necessary and sufficient conditions in various cases, focusing on general features and illustrating the method with some simple examples.
Unfortunately what counts as 'sufficiently good' is poorly understood, as are the limits of this approach. In this talk, I'll discuss a new framework (joint with Kevin Ford) which allows us to prove necessary and sufficient conditions in various cases, focusing on general features and illustrating the method with some simple examples.
Link to recording (YouTube)
Link to slides
Ilya Shkredov, Higher Sumsets and Energies in Additive Combinatorics and Number Theory
(Purdue University)
Ilya Shkredov, Higher Sumsets and Energies in Additive Combinatorics and Number Theory
(Purdue University)
Thursday, October 3, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 4, 2024 (1am AEDT, 4am NZDT)
Thursday, October 3, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 4, 2024 (1am AEDT, 4am NZDT)
Abstract: We provide an overview of the results obtained by the method of higher sumsets and higher energies to some problems of additive combinatorics (the sum—product phenomenon and incidence geometry, universality, additive decomposition, etc.), number theory (exponential sums over subgroups and Gauss sums, sums with multiplicative characters, the square—root barrier), Fourier analysis (uncertainty principle) and others. We will also discuss some perspectives for this approach.
Abstract: We provide an overview of the results obtained by the method of higher sumsets and higher energies to some problems of additive combinatorics (the sum—product phenomenon and incidence geometry, universality, additive decomposition, etc.), number theory (exponential sums over subgroups and Gauss sums, sums with multiplicative characters, the square—root barrier), Fourier analysis (uncertainty principle) and others. We will also discuss some perspectives for this approach.
Link to recording (YouTube)
Link to slides
Jordan Ellenberg, What does machine learning have to offer number theory?
(University of Wisconsin–Madison)
Jordan Ellenberg, What does machine learning have to offer number theory?
(University of Wisconsin–Madison)
Thursday, September 26, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 27, 2024 (1am AEST, 3am NZST)
Thursday, September 26, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 27, 2024 (1am AEST, 3am NZST)
Abstract: This is going to be a somewhat informal where I report on some of my own work, some work of others, and some stuff I’m learning about at the ongoing Harvard/CMSA workshop on machine learning in mathematics. I will focus on an outlook where the goal is not to reproduce or replace our central enterprise of writing proofs of theorems and understanding things, but rather on models for machine-human collaboration, where ML techniques are used to generate interesting hypotheses, examples, and ideas as a kind of force multiplier for traditional mathematicians. I’ll probably talk about cap sets, computing GCDs, murmurations, navigating Cayley graphs, and probably some other stuff besides! (Note: Oct 7-11 will be a number theory week at CMSA so any questions the audience wants to suggest we work on there are very welcome!)
Abstract: This is going to be a somewhat informal where I report on some of my own work, some work of others, and some stuff I’m learning about at the ongoing Harvard/CMSA workshop on machine learning in mathematics. I will focus on an outlook where the goal is not to reproduce or replace our central enterprise of writing proofs of theorems and understanding things, but rather on models for machine-human collaboration, where ML techniques are used to generate interesting hypotheses, examples, and ideas as a kind of force multiplier for traditional mathematicians. I’ll probably talk about cap sets, computing GCDs, murmurations, navigating Cayley graphs, and probably some other stuff besides! (Note: Oct 7-11 will be a number theory week at CMSA so any questions the audience wants to suggest we work on there are very welcome!)
Link to recording (YouTube)
Link to slides
Katherine E. Stange, The local-global conjecture for Apollonian circle packings is false
(University of Colorado, Boulder)
Katherine E. Stange, The local-global conjecture for Apollonian circle packings is false
(University of Colorado, Boulder)
CANCELLED
Thursday, September 19, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 20, 2024 (1am AEST, 3am NZST)
Thursday, September 19, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 20, 2024 (1am AEST, 3am NZST)
Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.
Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.
Mehtaab Sawhney, Primes of the form p^2 + nq^2
(Columbia University)
Mehtaab Sawhney, Primes of the form p^2 + nq^2
(Columbia University)
Thursday, September 12, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 13, 2024 (1am AEST, 3am NZST)
Thursday, September 12, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 13, 2024 (1am AEST, 3am NZST)
Abstract: Suppose that n is 0 or 4 mod 6. We show that there are infinitely many primes of the form p^2 + nq^2 with both p and q prime, and obtain an asymptotic for their number. In particular, when n = 4 we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec.
Abstract: Suppose that n is 0 or 4 mod 6. We show that there are infinitely many primes of the form p^2 + nq^2 with both p and q prime, and obtain an asymptotic for their number. In particular, when n = 4 we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec.
Joint w. Ben Green (Oxford)
Joint w. Ben Green (Oxford)
Link to recording (YouTube)
Link to slides
Rachel Newton, Evaluating the wild Brauer group
(King’s College London)
Rachel Newton, Evaluating the wild Brauer group
(King’s College London)
Thursday, September 5, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 6, 2024 (1am AEST, 3am NZST)
Thursday, September 5, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 6, 2024 (1am AEST, 3am NZST)
Abstract: The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer--Manin pairing cuts out a subset of the adelic points, called the Brauer--Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction to the existence of rational points on X. Computing the Brauer--Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer--Manin obstruction. This is joint work with Martin Bright.
Abstract: The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer--Manin pairing cuts out a subset of the adelic points, called the Brauer--Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction to the existence of rational points on X. Computing the Brauer--Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer--Manin obstruction. This is joint work with Martin Bright.
Link to recording (YouTube)
Chris Daw, Large Galois orbits under multiplicative degeneration
(University of Reading)
Chris Daw, Large Galois orbits under multiplicative degeneration
(University of Reading)
Thursday, June 27, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 28, 2024 (1am AEST, 3am NZST)
Thursday, June 27, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 28, 2024 (1am AEST, 3am NZST)
Abstract: The Pila-Zannier strategy is a powerful technique for proving results in unlikely intersections. In this talk, I will recall the Zilber-Pink conjecture for Shimura varieties and describe how Pila-Zannier works in this setting. I will highlight the most difficult outstanding obstacle to implementing the strategy — the so-called Large Galois Orbits conjecture — and I will explain recent progress towards this conjecture, building on the works of André and Bombieri. This is joint with Martin Orr (Manchester).
Abstract: The Pila-Zannier strategy is a powerful technique for proving results in unlikely intersections. In this talk, I will recall the Zilber-Pink conjecture for Shimura varieties and describe how Pila-Zannier works in this setting. I will highlight the most difficult outstanding obstacle to implementing the strategy — the so-called Large Galois Orbits conjecture — and I will explain recent progress towards this conjecture, building on the works of André and Bombieri. This is joint with Martin Orr (Manchester).
Link to recording (YouTube)
Max Wenqiang Xu, Real zeros of Fekete polynomials and positive definite characters
(Stanford University)
Max Wenqiang Xu, Real zeros of Fekete polynomials and positive definite characters
(Stanford University)
Thursday, June 20, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 21, 2024 (1am AEST, 3am NZST)
Thursday, June 20, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 21, 2024 (1am AEST, 3am NZST)
Abstract: In 1911, Fekete proposed the problem of studying how likely a Fekete polynomial has no real zeros in [0,1]. The work of Baker and Montgomery in 1989 qualitatively showed that Fekete polynomials without real zeros in [0,1] are rare. A closely related question is asking how likely a quadratic character has nonnegative partial sums at any stopping point. In a joint work (in progress) with Angelo and Soundararajan, we give a quantitative upper bound which is close to the conjectural bound.
Abstract: In 1911, Fekete proposed the problem of studying how likely a Fekete polynomial has no real zeros in [0,1]. The work of Baker and Montgomery in 1989 qualitatively showed that Fekete polynomials without real zeros in [0,1] are rare. A closely related question is asking how likely a quadratic character has nonnegative partial sums at any stopping point. In a joint work (in progress) with Angelo and Soundararajan, we give a quantitative upper bound which is close to the conjectural bound.
Link to recording (YouTube)
Link to slides
Philippe Michel, Mixed moments for Dirichlet L-functions
(EPFL)
Philippe Michel, Mixed moments for Dirichlet L-functions
(EPFL)
Thursday, June 13, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 14, 2024 (1am AEST, 3am NZST)
Thursday, June 13, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 14, 2024 (1am AEST, 3am NZST)
Abstract: In this talk we discuss the problem of evaluating somewhat exotic moments of Dirichlet L-functions of large modulus (called « mixed »).
Abstract: In this talk we discuss the problem of evaluating somewhat exotic moments of Dirichlet L-functions of large modulus (called « mixed »).
Namely moments of the shape
Namely moments of the shape
$$\sum_{\chi(q)} L(\chi^{a_1},1/2)\cdots L(\chi^a_k,1/2)$$
$$\sum_{\chi(q)} L(\chi^{a_1},1/2)\cdots L(\chi^a_k,1/2)$$
where $q$ is a growing prime and $a_i,\ 1\leq i\leq k$ are fixed integers (that are not necessarily equal nor equal to $\pm 1$).
where $q$ is a growing prime and $a_i,\ 1\leq i\leq k$ are fixed integers (that are not necessarily equal nor equal to $\pm 1$).
We will discuss some partial results focusing mainly on the case $k=2$ and $3$.
We will discuss some partial results focusing mainly on the case $k=2$ and $3$.
The techniques involved are non trivial bounds for solutions to monomial congruences equations as well as for averages of hyper-Kloosterman sums in short intervals.
The techniques involved are non trivial bounds for solutions to monomial congruences equations as well as for averages of hyper-Kloosterman sums in short intervals.
This is joint work with E. Fouvry, E. Kowalski and W. Sawin.
This is joint work with E. Fouvry, E. Kowalski and W. Sawin.
Link to recording (YouTube)
Link to slides
Dang-Khoa Nguyen, The Pólya-Carlson dichotomy for some dynamical zeta functions
(University of Calgary)
Dang-Khoa Nguyen, The Pólya-Carlson dichotomy for some dynamical zeta functions
(University of Calgary)
Thursday, June 6, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 7, 2024 (1am AEST, 3am NZST)
Thursday, June 6, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 7, 2024 (1am AEST, 3am NZST)
Abstract: Let $\theta$ be a map from a set $X$ to itself. Suppose that for $k\geq 1$, the number $N_k(\theta)$ of fixed points of the $k$-th fold iterate $\theta^k=\theta\circ\cdots\circ\theta$ is finite. Then we can define the dynamical or Artin-Mazur zeta function
Abstract: Let $\theta$ be a map from a set $X$ to itself. Suppose that for $k\geq 1$, the number $N_k(\theta)$ of fixed points of the $k$-th fold iterate $\theta^k=\theta\circ\cdots\circ\theta$ is finite. Then we can define the dynamical or Artin-Mazur zeta function
$$\zeta_{\theta}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{N_k(\theta)}{k}z^k\right).$$
$$\zeta_{\theta}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{N_k(\theta)}{k}z^k\right).$$
A complex power series with radius of convergence $R\in (0,\infty)$ is said to satisfy the P\'olya-Carlson dichotomy if it is either a rational function or it cannot be extended analytically beyond the disk of radius $R$.
A complex power series with radius of convergence $R\in (0,\infty)$ is said to satisfy the P\'olya-Carlson dichotomy if it is either a rational function or it cannot be extended analytically beyond the disk of radius $R$.
In this talk, we discuss the Pólya-Carlson dichotomy for the Artin-Mazur zeta functions of endomorphisms of tori and abelian varieties. This is from a joint work with Bell, Gunn, and Saunders and another with Baril Boudreau and Holmes.
In this talk, we discuss the Pólya-Carlson dichotomy for the Artin-Mazur zeta functions of endomorphisms of tori and abelian varieties. This is from a joint work with Bell, Gunn, and Saunders and another with Baril Boudreau and Holmes.
Link to recording (YouTube)
Link to slides
Rainer Dietmann, Longer gaps between values of binary quadratic forms
(Royal Holloway, University of London)
Rainer Dietmann, Longer gaps between values of binary quadratic forms
(Royal Holloway, University of London)
Thursday, May 30, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 31, 2024 (1am AEST, 3am NZST)
Thursday, May 30, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 31, 2024 (1am AEST, 3am NZST)
Abstract: It is not hard to show that there are infinitely many pairs of consecutive integers that are sums of two squares. The question about large gaps between sums of two squares is much more difficult. In this talk I want to report on recent joint work with Christian Elsholtz, Alexander Kalmynin, Sergei Konyagin and James Maynard which makes progress on this and related problems, in particular improving an old record of Richards from 1982.
Abstract: It is not hard to show that there are infinitely many pairs of consecutive integers that are sums of two squares. The question about large gaps between sums of two squares is much more difficult. In this talk I want to report on recent joint work with Christian Elsholtz, Alexander Kalmynin, Sergei Konyagin and James Maynard which makes progress on this and related problems, in particular improving an old record of Richards from 1982.
Link to recording (YouTube)
Link to slides
John Voight, The Bezout identity and norms from a quadratic extension
(Dartmouth College)
John Voight, The Bezout identity and norms from a quadratic extension
(Dartmouth College)
Thursday, May 23, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 24, 2024 (1am AEST, 3am NZST)
Thursday, May 23, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 24, 2024 (1am AEST, 3am NZST)
Abstract: Given coprime integers a,b, a classical identity provides integers u,v such that au-bv = 1. We consider refinements to this identity, where we ask that u,v are norms from a quadratic extension. This is joint work with Donald Cartwright and Xavier Roulleau.
Abstract: Given coprime integers a,b, a classical identity provides integers u,v such that au-bv = 1. We consider refinements to this identity, where we ask that u,v are norms from a quadratic extension. This is joint work with Donald Cartwright and Xavier Roulleau.
Link to recording (YouTube)
Link to slides
Nina Zubrilina, Murmurations of modular forms
(Princeton University)
Nina Zubrilina, Murmurations of modular forms
(Princeton University)
Thursday, May 16, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 17, 2024 (1am AEST, 3am NZST)
Thursday, May 16, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 17, 2024 (1am AEST, 3am NZST)
Abstract: In a recent machine learning-based study, He, Lee, Oliver, and Pozdnyakov observed an unexpected oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland later discovered that this bias extends to Dirichlet coefficients of other classes of L-functions when split by root number. In my talk, I will prove this bias for a family of holomorphic modular forms and for a family of Maass forms.
Abstract: In a recent machine learning-based study, He, Lee, Oliver, and Pozdnyakov observed an unexpected oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland later discovered that this bias extends to Dirichlet coefficients of other classes of L-functions when split by root number. In my talk, I will prove this bias for a family of holomorphic modular forms and for a family of Maass forms.
Link to recording (YouTube)
Jacob Tsimerman, Large Compact Subvarieties of A_g
(University of Toronto)
Jacob Tsimerman, Large Compact Subvarieties of A_g
(University of Toronto)
Thursday, May 9, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 10, 2024 (1am AEST, 3am NZST)
Thursday, May 9, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 10, 2024 (1am AEST, 3am NZST)
Abstract: (Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties A_g for any value of g. For g<16 the dimension is g-1, while for g>=16, it is determined by the larged dimensional compact shimura subvariety, which we determine. Our methods use functional transcendence theory.
Abstract: (Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties A_g for any value of g. For g<16 the dimension is g-1, while for g>=16, it is determined by the larged dimensional compact shimura subvariety, which we determine. Our methods use functional transcendence theory.
Link to recording (YouTube)
Link to slides
Ashwin Sah, Quasipolynomial bounds on the inverse theorem for the Gowers norms and applications
(MIT)
Ashwin Sah, Quasipolynomial bounds on the inverse theorem for the Gowers norms and applications
(MIT)
Thursday, May 2, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 3, 2024 (1am AEST, 3am NZST)
Thursday, May 2, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 3, 2024 (1am AEST, 3am NZST)
Abstract: Recent work, joint with James Leng and Mehtaab Sawhney, improves the so-called “inverse theorem” for the Gowers $U^{s+1}[N]$-norm which arises in the field of additive combinatorics in relation to Roth’s and Szemerédi’s theorems. I will explain how the field of higher-order Fourier analysis broadly extends Fourier methods and the circle method in number theory, and discuss implications of bounds for inverse theorems.
Abstract: Recent work, joint with James Leng and Mehtaab Sawhney, improves the so-called “inverse theorem” for the Gowers $U^{s+1}[N]$-norm which arises in the field of additive combinatorics in relation to Roth’s and Szemerédi’s theorems. I will explain how the field of higher-order Fourier analysis broadly extends Fourier methods and the circle method in number theory, and discuss implications of bounds for inverse theorems.
Link to recording (YouTube)
Link to slides
Theresa Anderson, Counting with new tools
(Carnegie Mellon University)
Theresa Anderson, Counting with new tools
(Carnegie Mellon University)
Thursday, April 25, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 26, 2024 (1am AEST, 3am NZST)
Thursday, April 25, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 26, 2024 (1am AEST, 3am NZST)
Abstract: Arithmetic statistics, or the counting of objects of algebraic interest, has seen a lot of development in the last twenty years. We will take a glimpse into just a few recent advances, with an emphasis on the wide interplay of new tools and techniques.
Abstract: Arithmetic statistics, or the counting of objects of algebraic interest, has seen a lot of development in the last twenty years. We will take a glimpse into just a few recent advances, with an emphasis on the wide interplay of new tools and techniques.
Link to recording (YouTube)
Link to slides
Uri Shapira, Distribution of conditional directional lattices
(Technion – Israel Institute of Technology)
Uri Shapira, Distribution of conditional directional lattices
(Technion – Israel Institute of Technology)
Thursday, April 18, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 19, 2024 (1am AEST, 3am NZST)
Thursday, April 18, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 19, 2024 (1am AEST, 3am NZST)
Abstract: Given an integral vector v in Euclidean n-space we project the standard lattice Z^n into the hyperplane orthogonal to v and obtain in this manner a "lattice of rank n-1" in that hyperplane, which is called "The directional lattice D(Z^n,v)".
Abstract: Given an integral vector v in Euclidean n-space we project the standard lattice Z^n into the hyperplane orthogonal to v and obtain in this manner a "lattice of rank n-1" in that hyperplane, which is called "The directional lattice D(Z^n,v)".
In this talk I will discuss results about the limit distribution of directional lattices as we let the vector v vary in some natural sets from a number theoretic point of view. These include, balls, spheres, non-compact quadratic surfaces, and integral vectors approximating an irrational line.
In this talk I will discuss results about the limit distribution of directional lattices as we let the vector v vary in some natural sets from a number theoretic point of view. These include, balls, spheres, non-compact quadratic surfaces, and integral vectors approximating an irrational line.
Link to recording (YouTube)
Link to slides
Yiannis Petridis, Counting and equidistribution
(University College London)
Yiannis Petridis, Counting and equidistribution
(University College London)
Thursday, April 11, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 12, 2024 (1am AEST, 3am NZST)
Thursday, April 11, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 12, 2024 (1am AEST, 3am NZST)
Abstract: I will discuss how counting orbits in hyperbolic spaces lead to interesting number theoretic problems. The counting problems (and the associated equidistribution) can be studied with various methods, and I will emphasize automorphic form techniques, originating in the work of H. Huber and studied extensively by A. Good. My collaborators in various aspects of this project are Chatzakos, Lekkas, Risager, and Voskou.
Abstract: I will discuss how counting orbits in hyperbolic spaces lead to interesting number theoretic problems. The counting problems (and the associated equidistribution) can be studied with various methods, and I will emphasize automorphic form techniques, originating in the work of H. Huber and studied extensively by A. Good. My collaborators in various aspects of this project are Chatzakos, Lekkas, Risager, and Voskou.
Link to recording (YouTube)
Link to slides
Dan Petersen, Moments of families of quadratic L-functions over function fields via homotopy theory
(Stockholm University)
Dan Petersen, Moments of families of quadratic L-functions over function fields via homotopy theory
(Stockholm University)
Thursday, April 4, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 5, 2024 (2am AEDT, 5am NZDT)
Thursday, April 4, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 5, 2024 (2am AEDT, 5am NZDT)
Abstract: This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. Our work concerns the CFKRS predictions in the case of the quadratic family over function fields, i.e. the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability with these coefficients. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.
Abstract: This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. Our work concerns the CFKRS predictions in the case of the quadratic family over function fields, i.e. the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability with these coefficients. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.
Link to recording (YouTube)
Link to slides
Wouter Castryck, The isogeny interpolation problem
(KU Leuven)
Wouter Castryck, The isogeny interpolation problem
(KU Leuven)
Thursday, March 28, 2024 (10am PDT, 1pm EDT, 5pm GMT, 6pm CET, 7pm Israel Standard Time, 10:30pm Indian Standard Time)
Friday, March 29, 2024 (1am CST, 4am AEDT, 6am NZDT)
Thursday, March 28, 2024 (10am PDT, 1pm EDT, 5pm GMT, 6pm CET, 7pm Israel Standard Time, 10:30pm Indian Standard Time)
Friday, March 29, 2024 (1am CST, 4am AEDT, 6am NZDT)
Abstract: It is easy to prove that a degree-d isogeny f between two elliptic curves E and E' is completely determined by the images of any 4d + 1 points. In this talk we will study the algorithmic problem of evaluating f at a given point P on E, merely upon input of such "interpolation data". In case the interpolation points generate a group containing E[N] such that N^2 > 4d is smooth and coprime to d and the field characteristic, this problem was solved in 2022 by Robert, in the context of breaking SIKE (= SIDH), a former candidate for post-quantum key exchange that had advanced to the final stage of a standardization effort run by the National Institute of Standards and Technology. We will discuss this solution, and then show how to address more general instances of the isogeny interpolation problem, while also publicizing some unsolved cases.
Abstract: It is easy to prove that a degree-d isogeny f between two elliptic curves E and E' is completely determined by the images of any 4d + 1 points. In this talk we will study the algorithmic problem of evaluating f at a given point P on E, merely upon input of such "interpolation data". In case the interpolation points generate a group containing E[N] such that N^2 > 4d is smooth and coprime to d and the field characteristic, this problem was solved in 2022 by Robert, in the context of breaking SIKE (= SIDH), a former candidate for post-quantum key exchange that had advanced to the final stage of a standardization effort run by the National Institute of Standards and Technology. We will discuss this solution, and then show how to address more general instances of the isogeny interpolation problem, while also publicizing some unsolved cases.
Link to recording (YouTube)
Link to slides
Javier Fresán, E-functions and Geometry
(Sorbonne University)
Javier Fresán, E-functions and Geometry
(Sorbonne University)
Thursday, March 21, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 22, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 21, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 22, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: E-functions are power series which solve a differential equation and whose coefficients are algebraic numbers that satisfy certain growth conditions of arithmetic nature. They were introduced in Siegel's 1929 memoir on the applications of diophantine approximation with the goal of generalising the Hermite--Lindemann--Weierstrass theorem about the transcendence of the values of the exponential function at algebraic arguments. Besides the exponential, standard examples include the Bessel function and confluent hypergeometric series. After briefly surveying on the history of E-functions, I will present a joint work in progress with Peter Jossen where we prove that exponential period functions provide us with a rich geometric source of E-functions. The easiest examples, attached to polynomials of degree 4, already allowed us a couple of years ago to exhibit some E-functions which are not polynomial expressions in hypergeometric series, thus solving one of the problems in Siegel's original paper.
Abstract: E-functions are power series which solve a differential equation and whose coefficients are algebraic numbers that satisfy certain growth conditions of arithmetic nature. They were introduced in Siegel's 1929 memoir on the applications of diophantine approximation with the goal of generalising the Hermite--Lindemann--Weierstrass theorem about the transcendence of the values of the exponential function at algebraic arguments. Besides the exponential, standard examples include the Bessel function and confluent hypergeometric series. After briefly surveying on the history of E-functions, I will present a joint work in progress with Peter Jossen where we prove that exponential period functions provide us with a rich geometric source of E-functions. The easiest examples, attached to polynomials of degree 4, already allowed us a couple of years ago to exhibit some E-functions which are not polynomial expressions in hypergeometric series, thus solving one of the problems in Siegel's original paper.
Link to recording (YouTube)
Link to slides
Marc Munsch, Two tales on quadratic character sums
(Jean Monnet University)
Marc Munsch, Two tales on quadratic character sums
(Jean Monnet University)
Thursday, March 14, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 15, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 14, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 15, 2024 (12am CST, 3am AEDT, 5am NZDT)
Link to recording (YouTube)
Link to slides
Vesselin Dimitrov, The next case after Apéry on mixed Tate periods
(Caltech)
Vesselin Dimitrov, The next case after Apéry on mixed Tate periods
(Caltech)
Thursday, March 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 8, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 8, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will introduce a method, joint with Frank Calegari and Yunqing Tang, for proving linear independence results and effective bad approximability measures. It is an outgrowth of our previous joint work on the so-called "unbounded denominators conjecture," which was in some sense an application of transcendental number theory to modular forms theory, with the key step being to prove sufficiently sharp $\mathbb{Q}(x)$-linear dimension bounds on certain spaces of algebraic functions. This time, we step into the wilder realm of G-functions with infinite monodromy, and devise holonomy bounds fine enough to prove the linear independence of two certain Dirichlet L-function values, a result that, in the realm of mixed Tate periods, can be considered as the next-simplest case after Apery's proof of the irrationality of $\zeta(3)$ (excluding the cases that reduce to the Hermite--Lindemann theorem or the Gelfond--Baker theorem on linear forms in logarithms). One key input turns out to be the classical Shidlovsky lemma on functional bad approximability, the point Siegel missed for three decades to complete his theory of algebraic relations among special values of E-functions.
Abstract: I will introduce a method, joint with Frank Calegari and Yunqing Tang, for proving linear independence results and effective bad approximability measures. It is an outgrowth of our previous joint work on the so-called "unbounded denominators conjecture," which was in some sense an application of transcendental number theory to modular forms theory, with the key step being to prove sufficiently sharp $\mathbb{Q}(x)$-linear dimension bounds on certain spaces of algebraic functions. This time, we step into the wilder realm of G-functions with infinite monodromy, and devise holonomy bounds fine enough to prove the linear independence of two certain Dirichlet L-function values, a result that, in the realm of mixed Tate periods, can be considered as the next-simplest case after Apery's proof of the irrationality of $\zeta(3)$ (excluding the cases that reduce to the Hermite--Lindemann theorem or the Gelfond--Baker theorem on linear forms in logarithms). One key input turns out to be the classical Shidlovsky lemma on functional bad approximability, the point Siegel missed for three decades to complete his theory of algebraic relations among special values of E-functions.
This is all a joint work with Frank Calegari and Yunqing Tang.
This is all a joint work with Frank Calegari and Yunqing Tang.
Paul Pollack, Stretching, the truth about nonunique factorization
(University of Georgia)
Paul Pollack, Stretching, the truth about nonunique factorization
(University of Georgia)
Thursday, February 29, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 1, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 29, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 1, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Number theorists learn at their mother's knee that unique factorization fails in \Z[\sqrt{-5}]. Less well-known is that \Z[\sqrt{-5}] exhibits only a "half-failure" of unique factorization: while two factorizations into irreducibles of the same element need not agree up to unit factors, their lengths (number of factors) does always agree. This is a special case of a 1960 result of Leonard Carlitz. I will discuss offshoots of Carlitz's theorem. Particular attention will be paid to certain questions of Coykendall regarding "elasticity" of orders in quadratic number fields.
Abstract: Number theorists learn at their mother's knee that unique factorization fails in \Z[\sqrt{-5}]. Less well-known is that \Z[\sqrt{-5}] exhibits only a "half-failure" of unique factorization: while two factorizations into irreducibles of the same element need not agree up to unit factors, their lengths (number of factors) does always agree. This is a special case of a 1960 result of Leonard Carlitz. I will discuss offshoots of Carlitz's theorem. Particular attention will be paid to certain questions of Coykendall regarding "elasticity" of orders in quadratic number fields.
Link to recording (YouTube)
Link to slides
Arul Shankar, Secondary terms in the first moment of the 2-Selmer groups of elliptic curves
(University of Toronto)
Arul Shankar, Secondary terms in the first moment of the 2-Selmer groups of elliptic curves
(University of Toronto)
Thursday, February 22, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 23, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 22, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 23, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A web of interrelated conjectures (due to work of Goldfeld, Katz--Sarnak, Poonen-Rains, Bhargava--Kane--Lenstra--Poonen--Rains) predict the distributions of ranks and Selmer groups of elliptic curves over Q. These conjectures predict that the average rank of elliptic curves is 1/2. Furthermore, it is known (due to Bhargava and myself) that the average size of the 2-Selmer group of elliptic curves is 3 (when the family of all elliptic curves is ordered by (naive) height).
Abstract: A web of interrelated conjectures (due to work of Goldfeld, Katz--Sarnak, Poonen-Rains, Bhargava--Kane--Lenstra--Poonen--Rains) predict the distributions of ranks and Selmer groups of elliptic curves over Q. These conjectures predict that the average rank of elliptic curves is 1/2. Furthermore, it is known (due to Bhargava and myself) that the average size of the 2-Selmer group of elliptic curves is 3 (when the family of all elliptic curves is ordered by (naive) height).
On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find both a larger average rank as well as a smaller average size of the 2-Selmer group in the data. In this talk, we will discuss joint work with Takashi Taniguchi, in which we give a possible theoretical explanation for deviation of the data on 2-Selmer groups from the predicted distribution, namely, the existence of a secondary term.
On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find both a larger average rank as well as a smaller average size of the 2-Selmer group in the data. In this talk, we will discuss joint work with Takashi Taniguchi, in which we give a possible theoretical explanation for deviation of the data on 2-Selmer groups from the predicted distribution, namely, the existence of a secondary term.
Link to recording (YouTube)
Link to slides
Damaris Schindler, Density of rational points near manifolds
(University of Göttingen)
Damaris Schindler, Density of rational points near manifolds
(University of Göttingen)
Thursday, February 15, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 16, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 15, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 16, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.
Abstract: Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.
Link to recording (YouTube)
Victor Y. Wang, Sums of three cubes over a function field
(IST Austria)
Victor Y. Wang, Sums of three cubes over a function field
(IST Austria)
Thursday, February 8, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 9, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 8, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 9, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will talk about joint work with Tim Browning and Jakob Glas on producing sums of three cubes over a function field, assuming a q-restricted form of the Ratios Conjecture for a geometric family of L-functions. If time permits, I may also discuss some recent developments in homological stability that could help to resolve this q-restricted Ratios Conjecture.
Abstract: I will talk about joint work with Tim Browning and Jakob Glas on producing sums of three cubes over a function field, assuming a q-restricted form of the Ratios Conjecture for a geometric family of L-functions. If time permits, I may also discuss some recent developments in homological stability that could help to resolve this q-restricted Ratios Conjecture.
Link to slides (Note by the organizers: the final slide was added after the talk following a comment of Trevor Wooley.)
Link to slides (Note by the organizers: the final slide was added after the talk following a comment of Trevor Wooley.)
Link to recording (YouTube)
Akshat Mudgal, Recent progress towards the sum–product conjecture and related problems
(University of Oxford)
Akshat Mudgal, Recent progress towards the sum–product conjecture and related problems
(University of Oxford)
Thursday, February 1, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 2, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 1, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 2, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: An important open problem in combinatorial number theory is the Erdös–Szemerédi sum–product conjecture, which suggests that for any positive integers s, N, and for any set A of N integers, either there are many s-fold sums of the form a_1 + … + a_s or there are many s-fold products of the form a_1…a_s. While this remains wide open, various generalisations of this problem have been considered more recently, including the question of finding large additive and multiplicative Sidon sets in arbitrary sets of integers as well as studying the so-called low energy decompositions.
Abstract: An important open problem in combinatorial number theory is the Erdös–Szemerédi sum–product conjecture, which suggests that for any positive integers s, N, and for any set A of N integers, either there are many s-fold sums of the form a_1 + … + a_s or there are many s-fold products of the form a_1…a_s. While this remains wide open, various generalisations of this problem have been considered more recently, including the question of finding large additive and multiplicative Sidon sets in arbitrary sets of integers as well as studying the so-called low energy decompositions.
In this talk, I will outline some recent progress towards the above questions, as well as highlight how these connect very naturally to other key conjectures in additive combinatorics.
In this talk, I will outline some recent progress towards the above questions, as well as highlight how these connect very naturally to other key conjectures in additive combinatorics.
Link to recording (YouTube)
Thomas Tucker, Tits and Borel type theorems for preperiodic points of finite morphisms
(University of Rochester)
Thomas Tucker, Tits and Borel type theorems for preperiodic points of finite morphisms
(University of Rochester)
Thursday, January 25, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 26, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 25, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 26, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We pose a general question: Given a finitely generated semigroup S of finite morphisms from a variety to itself, what can one say about how the structure of the semigroup is connected to the relationship between the preperiodic points of the elements of S? When S consists of polarized morphisms, we can give a fairly simple answer to this question using Tate's limiting procedure for Weil and Moriwaki heights. We formulate some conjectures that generalize this answer and prove some results relating to these conjectures.
Abstract: We pose a general question: Given a finitely generated semigroup S of finite morphisms from a variety to itself, what can one say about how the structure of the semigroup is connected to the relationship between the preperiodic points of the elements of S? When S consists of polarized morphisms, we can give a fairly simple answer to this question using Tate's limiting procedure for Weil and Moriwaki heights. We formulate some conjectures that generalize this answer and prove some results relating to these conjectures.
Link to recording (YouTube)
Oded Regev, An Efficient Quantum Factoring Algorithm
(Courant Institute of Mathematical Sciences)
Oded Regev, An Efficient Quantum Factoring Algorithm
(Courant Institute of Mathematical Sciences)
Thursday, January 18, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 19, 2024 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 18, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 19, 2024 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We show that n-bit integers can be factorized by independently running a quantum circuit with \tilde{O}(n^{3/2}) gates for \sqrt{n}+4 times, and then using polynomial-time classical post-processing. In contrast, Shor's algorithm requires circuits with \tilde{O}(n^2) gates. The
Abstract: We show that n-bit integers can be factorized by independently running a quantum circuit with \tilde{O}(n^{3/2}) gates for \sqrt{n}+4 times, and then using polynomial-time classical post-processing. In contrast, Shor's algorithm requires circuits with \tilde{O}(n^2) gates. The
correctness of our algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.
correctness of our algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.
No background in quantum computation will be assumed.
No background in quantum computation will be assumed.
Link to recording (YouTube)
Zeev Rudnick, A talk in honor of Peter Sarnak's 70th birthday
(Tel Aviv University)
Zeev Rudnick, A talk in honor of Peter Sarnak's 70th birthday
(Tel Aviv University)
Celebrating Peter Sarnak's 70th birthday
Special Chair: Alex Kontorovich (Rutgers University)
Special Chair: Alex Kontorovich (Rutgers University)
Thursday, December 21, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 22, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 21, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 22, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will give selected highlights of Peter Sarnak's works on automorphic forms and some of the outstanding problems remaining.
Abstract: I will give selected highlights of Peter Sarnak's works on automorphic forms and some of the outstanding problems remaining.
Link to recording (YouTube)
Djordje Milićević, Beyond the spherical sup-norm problem
(Bryn Mawr College)
Djordje Milićević, Beyond the spherical sup-norm problem
(Bryn Mawr College)
Thursday, December 14, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 15, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 14, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 15, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The sup-norm problem on arithmetic Riemannian manifolds occupies a prominent place at the intersection of harmonic analysis, number theory, and quantum mechanics. It asks about the sup-norm of L^2-normalized joint eigenfunctions of invariant differential operators and Hecke operators — that is, automorphic forms — most classically in terms of their Laplace eigenvalues (as in the QUE problem for high-energy eigenstates), but also in terms of the volume of the manifold and other parameters.
Abstract: The sup-norm problem on arithmetic Riemannian manifolds occupies a prominent place at the intersection of harmonic analysis, number theory, and quantum mechanics. It asks about the sup-norm of L^2-normalized joint eigenfunctions of invariant differential operators and Hecke operators — that is, automorphic forms — most classically in terms of their Laplace eigenvalues (as in the QUE problem for high-energy eigenstates), but also in terms of the volume of the manifold and other parameters.
In this talk, we will motivate the sup-norm problem and then describe our results, joint with Blomer, Harcos, and Maga, which for the first time solve it for non-spherical Maass forms of an increasing dimension of the associated K-type, on an arithmetic quotient of G=SL(2,C), with K=SU(2). We combine representation theory, spectral analysis, and Diophantine arguments, developing new Paley-Wiener theory for G and sharp estimates on spherical trace functions of arbitrary K-type on the way to a novel counting problem of Hecke correspondences close to various special submanifolds of G.
In this talk, we will motivate the sup-norm problem and then describe our results, joint with Blomer, Harcos, and Maga, which for the first time solve it for non-spherical Maass forms of an increasing dimension of the associated K-type, on an arithmetic quotient of G=SL(2,C), with K=SU(2). We combine representation theory, spectral analysis, and Diophantine arguments, developing new Paley-Wiener theory for G and sharp estimates on spherical trace functions of arbitrary K-type on the way to a novel counting problem of Hecke correspondences close to various special submanifolds of G.
Link to recording (YouTube)
Misha Rudnev, The sum-product problem for integers with few prime factors
(University of Bristol)
Misha Rudnev, The sum-product problem for integers with few prime factors
(University of Bristol)
Thursday, December 7, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 8, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 7, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 8, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: It was asked by Szemerédi if the known sum-product estimates can be improved for a set of N integers under the constraint that each integer has a small number of prime factors. We prove, if the maximum number of prime factors for each integer is sub-logarithmic in N, the sum-product exponent 5/3-o(1).
Abstract: It was asked by Szemerédi if the known sum-product estimates can be improved for a set of N integers under the constraint that each integer has a small number of prime factors. We prove, if the maximum number of prime factors for each integer is sub-logarithmic in N, the sum-product exponent 5/3-o(1).
This becomes a corollary of an additive energy versus the product set cardinality estimate, which turns out to be the best possible.
This becomes a corollary of an additive energy versus the product set cardinality estimate, which turns out to be the best possible.
It is based on a scheme of Burkholder-Gundy-Davis martingale square function inequalities in p-adic scales, followed by an application of a variant of the Schmidt subspace theorem.
It is based on a scheme of Burkholder-Gundy-Davis martingale square function inequalities in p-adic scales, followed by an application of a variant of the Schmidt subspace theorem.
Link to recording (YouTube)
Wei Zhang, Diagonal cycles: some results and conjectures
(MIT)
Wei Zhang, Diagonal cycles: some results and conjectures
(MIT)
Thursday, November 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 1, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 1, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Algebraic cycles are among the most fundamental mathematical objects. I will discuss a class of special algebraic cycles related to the diagonal cycle, including the Gross-Schoen cycle (the small diagonal) on the triple product of a curve, the arithmetic diagonal cycle appearing in the Gan-Gross-Prasad conjecture, as well as the Fourier-Jacobi cycle defined by Yifeng Liu.
Abstract: Algebraic cycles are among the most fundamental mathematical objects. I will discuss a class of special algebraic cycles related to the diagonal cycle, including the Gross-Schoen cycle (the small diagonal) on the triple product of a curve, the arithmetic diagonal cycle appearing in the Gan-Gross-Prasad conjecture, as well as the Fourier-Jacobi cycle defined by Yifeng Liu.
Link to recording (YouTube)
Anke Pohl, Period functions for vector-valued automorphic functions via dynamics and cohomology
(University of Bremen)
Anke Pohl, Period functions for vector-valued automorphic functions via dynamics and cohomology
(University of Bremen)
Thursday, November 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Vector-valued automorphic functions, or generalized automorphic functions, occur naturally in many areas, most notably in spectral theory, number theory and mathematical physics. Already Selberg promoted the idea to investigate vector-valued automorphic functions alongside their classical relatives and to exploit their interaction in order to understand their properties. While during the last decades the focus has been on automorphic functions equivariant with regard to unitary representations, the investigations recently turned to non-unitary representations as well. I will report on the status of an ongoing project to investigate simultaneously unitarily and non-unitarily equivariant automorphic functions with a view towards period functions and a classical-quantum correspondence by means of dynamics (transfer operator methods) and cohomology theory. This is joint work with R. Bruggeman.
Abstract: Vector-valued automorphic functions, or generalized automorphic functions, occur naturally in many areas, most notably in spectral theory, number theory and mathematical physics. Already Selberg promoted the idea to investigate vector-valued automorphic functions alongside their classical relatives and to exploit their interaction in order to understand their properties. While during the last decades the focus has been on automorphic functions equivariant with regard to unitary representations, the investigations recently turned to non-unitary representations as well. I will report on the status of an ongoing project to investigate simultaneously unitarily and non-unitarily equivariant automorphic functions with a view towards period functions and a classical-quantum correspondence by means of dynamics (transfer operator methods) and cohomology theory. This is joint work with R. Bruggeman.
Link to recording (YouTube)
Henri Darmon, Explicit class field theory and orthogonal groups
(McGill University)
Henri Darmon, Explicit class field theory and orthogonal groups
(McGill University)
Thursday, November 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Essentially all abelian extensions of the rational numbers or of a quadratic imaginary field can be generated by special values of the exponential function or of the modular j-function at explicit arguments in the ground field. Describing the mathematical objects which could play the role of trigonometric and modular functions in generating class fields of more general base fields is the stated goal of explicit class field theory. Around 5 years ago Jan Vonk and I proposed a framework in which class fields of real quadratic fields can be generated from the special values of certain “rigid meromorphic cocycles” at real quadratic arguments. Without delving into the details of this framework, I will present some simple concrete consequences of it in settings where the base field is totally real, and explain how they can be proved. The more general statements rest on (but do not require the full force of) the notion of rigid meromorphic cocycles for orthogonal groups of signature (r,r) described in joint work with Lennart Gehrmann and Mike Lipnowski, and are also inspired by the calculations in Romain Branchereau’s PhD thesis. (Joint with Jan Vonk)
Abstract: Essentially all abelian extensions of the rational numbers or of a quadratic imaginary field can be generated by special values of the exponential function or of the modular j-function at explicit arguments in the ground field. Describing the mathematical objects which could play the role of trigonometric and modular functions in generating class fields of more general base fields is the stated goal of explicit class field theory. Around 5 years ago Jan Vonk and I proposed a framework in which class fields of real quadratic fields can be generated from the special values of certain “rigid meromorphic cocycles” at real quadratic arguments. Without delving into the details of this framework, I will present some simple concrete consequences of it in settings where the base field is totally real, and explain how they can be proved. The more general statements rest on (but do not require the full force of) the notion of rigid meromorphic cocycles for orthogonal groups of signature (r,r) described in joint work with Lennart Gehrmann and Mike Lipnowski, and are also inspired by the calculations in Romain Branchereau’s PhD thesis. (Joint with Jan Vonk)
Link to recording (YouTube)
Jens Marklof, Smallest denominators
(University of Bristol)
Jens Marklof, Smallest denominators
(University of Bristol)
Thursday, November 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: If we partition the unit interval into 3000 equal subintervals and take the smallest denominator amongst all rational points in each subinterval, what can we say about the distribution of those 3000 denominators? I will discuss this and related questions, its connection with Farey statistics and random lattices. In particular, I will report on higher dimensional versions of a recent proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305] on the convergence of the expectation value of the above distribution, as well as closely related work by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405--1413]. In fact, we will uncover the full distribution and prove convergence of more moments than just the expectation value. (This I believe was previously not known even in one dimension.) We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130--149] as well as new results on pigeonhole statistics.
Abstract: If we partition the unit interval into 3000 equal subintervals and take the smallest denominator amongst all rational points in each subinterval, what can we say about the distribution of those 3000 denominators? I will discuss this and related questions, its connection with Farey statistics and random lattices. In particular, I will report on higher dimensional versions of a recent proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305] on the convergence of the expectation value of the above distribution, as well as closely related work by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405--1413]. In fact, we will uncover the full distribution and prove convergence of more moments than just the expectation value. (This I believe was previously not known even in one dimension.) We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130--149] as well as new results on pigeonhole statistics.
Link to recording (YouTube)
Joseph H. Silverman, Field of Moduli and Fields of Definition in Arithmetic Geometry and Arithmetic Dynamics
(Brown University)
Joseph H. Silverman, Field of Moduli and Fields of Definition in Arithmetic Geometry and Arithmetic Dynamics
(Brown University)
Thursday, November 2, 2023 (8am PDT, 11am EDT, 3pm GMT, 4pm CET, 5pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, November 3, 2023 (2am AEDT, 4am NZDT)
Thursday, November 2, 2023 (8am PDT, 11am EDT, 3pm GMT, 4pm CET, 5pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, November 3, 2023 (2am AEDT, 4am NZDT)
Abstract: Let X/Qbar be an algebraic variety defined over the field of algebraic numbers. We say that a number field K is a field of definition (FOD) for X if there is a variety Y/K such that Y is Qbar-isomorphic to X.
Abstract: Let X/Qbar be an algebraic variety defined over the field of algebraic numbers. We say that a number field K is a field of definition (FOD) for X if there is a variety Y/K such that Y is Qbar-isomorphic to X.
The field of moduli (FOM) of X is the fixed field of
The field of moduli (FOM) of X is the fixed field of
{ s \in G_Q : s(X) is Qbar-isomorphic to X }.
{ s \in G_Q : s(X) is Qbar-isomorphic to X }.
It is easy to check that every FOD for X contains the FOM of X, but there are many situations where the FOM of X is not a FOD. I will briefly discuss the FOM versus FOD problem in the classical case of abelian varieties, and then turn to the the analogous question for morphisms f : P^N --> P^N defined over Qbar, where two maps are (dynamically) isomorphic if they are conjugate by a linear fractional transformation. I will describe what is known for N=1, including examples of maps for which the FOM is not an FOD. I will then discuss recent results for higher dimensional projective spaces in which we show that every map f has a FOD whose degree over its FOM is bounded by a function depending only on N and deg(f). (Joint work with John Doyle.)
It is easy to check that every FOD for X contains the FOM of X, but there are many situations where the FOM of X is not a FOD. I will briefly discuss the FOM versus FOD problem in the classical case of abelian varieties, and then turn to the the analogous question for morphisms f : P^N --> P^N defined over Qbar, where two maps are (dynamically) isomorphic if they are conjugate by a linear fractional transformation. I will describe what is known for N=1, including examples of maps for which the FOM is not an FOD. I will then discuss recent results for higher dimensional projective spaces in which we show that every map f has a FOD whose degree over its FOM is bounded by a function depending only on N and deg(f). (Joint work with John Doyle.)
Link to recording (YouTube)
Ananth Shankar, Canonical heights on Shimura varieties and the Andre-Oort conjecture
(University of Wisconsin, Madison)
Ananth Shankar, Canonical heights on Shimura varieties and the Andre-Oort conjecture
(University of Wisconsin, Madison)
Thursday, October 26, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 27, 2023 (2am AEDT, 4am NZDT)
Thursday, October 26, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 27, 2023 (2am AEDT, 4am NZDT)
Abstract: Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. Binyamini-Schmidt-Yafaev build on work of Binyamini to reduce the Andre-Oort conjecture to establishing height bounds on special points. I will describe joint work with Jonathan Pila and Jacob Tsimerman where we establish these height bounds, and therefore prove the Andre Oort conjecture in full generality.
Abstract: Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. Binyamini-Schmidt-Yafaev build on work of Binyamini to reduce the Andre-Oort conjecture to establishing height bounds on special points. I will describe joint work with Jonathan Pila and Jacob Tsimerman where we establish these height bounds, and therefore prove the Andre Oort conjecture in full generality.
Link to recording (YouTube)
Timothy Browning, When is a random Diophantine equation soluble over $\mathbb{Q}_p$ for all $p$?
(IST Austria)
Timothy Browning, When is a random Diophantine equation soluble over $\mathbb{Q}_p$ for all $p$?
(IST Austria)
Thursday, October 19, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 20, 2023 (2am AEDT, 4am NZDT)
Thursday, October 19, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 20, 2023 (2am AEDT, 4am NZDT)
Abstract: The question in the title is of growing importance in number theory and represents a more tractable staging post than the question of solubility over $\mathbb{Q}$. I'll describe the landscape for various families of varieties, which can be interpreted as a more delicate version of Manin's conjecture, in which one counts rational points of bounded height which lie in the image of adelic points under a morphism. This leads to more subtle asymptotic behaviours and depends intimately on the geometry of the morphism. This is joint work with Julian Lyczak, Roman Sarapin and Arne Smeets.
Abstract: The question in the title is of growing importance in number theory and represents a more tractable staging post than the question of solubility over $\mathbb{Q}$. I'll describe the landscape for various families of varieties, which can be interpreted as a more delicate version of Manin's conjecture, in which one counts rational points of bounded height which lie in the image of adelic points under a morphism. This leads to more subtle asymptotic behaviours and depends intimately on the geometry of the morphism. This is joint work with Julian Lyczak, Roman Sarapin and Arne Smeets.
Link to recording (YouTube)
David Masser, Some new elliptic integrals
(University of Basel)
David Masser, Some new elliptic integrals
(University of Basel)
Thursday, October 12, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 13, 2023 (2am AEDT, 4am NZDT)
Thursday, October 12, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 13, 2023 (2am AEDT, 4am NZDT)
Link to recording (YouTube)
Isabel Vogt, Brauer--Manin obstructions requiring arbitrarily many Brauer classes
(Brown University)
Isabel Vogt, Brauer--Manin obstructions requiring arbitrarily many Brauer classes
(Brown University)
Thursday, October 5, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 6, 2023 (2am AEDT, 4am NZDT)
Thursday, October 5, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 6, 2023 (2am AEDT, 4am NZDT)
Abstract: A fundamental problem in the arithmetic of varieties over global fields is to determine whether they have a rational point. As a first effective step, one can check that a variety has local points for each place. However, this is not enough, as many classes of varieties are known to fail this local-global principle. The Brauer–Manin obstruction to the local-global principle for rational points is captured by elements of the Brauer group. On a projective variety, any Brauer–Manin obstruction is captured by a finite subgroup of the Brauer group. I will explain joint work that shows that this subgroup can require arbitrarily many generators. This is joint with J. Berg, C. Pagano, B. Poonen, M. Stoll, N. Triantafillou and B. Viray.
Abstract: A fundamental problem in the arithmetic of varieties over global fields is to determine whether they have a rational point. As a first effective step, one can check that a variety has local points for each place. However, this is not enough, as many classes of varieties are known to fail this local-global principle. The Brauer–Manin obstruction to the local-global principle for rational points is captured by elements of the Brauer group. On a projective variety, any Brauer–Manin obstruction is captured by a finite subgroup of the Brauer group. I will explain joint work that shows that this subgroup can require arbitrarily many generators. This is joint with J. Berg, C. Pagano, B. Poonen, M. Stoll, N. Triantafillou and B. Viray.
Link to recording (YouTube)
Link to slides
Holly Krieger, A transcendental birational dynamical degree
(University of Cambridge)
Holly Krieger, A transcendental birational dynamical degree
(University of Cambridge)
Thursday, September 28, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 29, 2023 (1am AEST, 4am NZDT)
Thursday, September 28, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 29, 2023 (1am AEST, 4am NZDT)
Abstract: In the study of a discrete dynamical system defined by polynomials, we wish to understand the integer sequence formed by the degrees of the iterates of the map: examples of such a sequence include the Fibonacci and other integer linear recurrence sequences, but not all examples satisfy a finite recurrence. The growth of this sequence is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for a birational map of projective space, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.
Abstract: In the study of a discrete dynamical system defined by polynomials, we wish to understand the integer sequence formed by the degrees of the iterates of the map: examples of such a sequence include the Fibonacci and other integer linear recurrence sequences, but not all examples satisfy a finite recurrence. The growth of this sequence is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for a birational map of projective space, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.
Link to recording (YouTube)
Link to slides
Kaisa Matomäki, Detecting primes in multiplicatively structured sequences
(University of Turku)
Kaisa Matomäki, Detecting primes in multiplicatively structured sequences
(University of Turku)
Thursday, September 21, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 22, 2023 (1am AEST, 4am NZDT)
Thursday, September 21, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 22, 2023 (1am AEST, 4am NZDT)
Abstract: I will discuss a new sieve set-up which allows one to find prime numbers in sequences that have a suitable multiplicative structure and good "type I information". Among other things, the method gives a new L-function free proof of Linnik's theorem concerning the least prime in an arithmetic progression. The talk is based on on-going joint work with Jori Merikoski and Joni Teräväinen.
Abstract: I will discuss a new sieve set-up which allows one to find prime numbers in sequences that have a suitable multiplicative structure and good "type I information". Among other things, the method gives a new L-function free proof of Linnik's theorem concerning the least prime in an arithmetic progression. The talk is based on on-going joint work with Jori Merikoski and Joni Teräväinen.
Link to recording (YouTube)
Link to slides
Andrew Sutherland, Murmurations of arithmetic L-functions
(MIT)
Andrew Sutherland, Murmurations of arithmetic L-functions
(MIT)
Thursday, September 14, 2023 (9am PDT, 12pm EDT, 5pm BST, 6pm CEST, 7pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, September 15, 2023 (12am CST, 2am AEST, 4am NZST)
Thursday, September 14, 2023 (9am PDT, 12pm EDT, 5pm BST, 6pm CEST, 7pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, September 15, 2023 (12am CST, 2am AEST, 4am NZST)
Abstract: While conducting a series of number-theoretic machine learning experiments last year, He, Lee, Oliver, and Pozdnyakov noticed a curious oscillation in the averages of Frobenius traces of elliptic curves over Q. If one computes the average value of a_p(E) for E/Q of fixed rank with conductor in a short interval, as p increases the average oscillates with a decaying frequency determined by the conductor. That the rank influences the distribution of Frobenius traces has long been known (indeed, this was the impetus for the experiments that led to the conjecture of Birch and Swinnerton-Dyer), but these oscillations do not appear to have been noticed previously. This may be due in part to the critical role played by the conductor; in arithmetic statistics it is common to order elliptic curves E/Q by naive height rather than conductor, but doing so obscures these oscillations.
Abstract: While conducting a series of number-theoretic machine learning experiments last year, He, Lee, Oliver, and Pozdnyakov noticed a curious oscillation in the averages of Frobenius traces of elliptic curves over Q. If one computes the average value of a_p(E) for E/Q of fixed rank with conductor in a short interval, as p increases the average oscillates with a decaying frequency determined by the conductor. That the rank influences the distribution of Frobenius traces has long been known (indeed, this was the impetus for the experiments that led to the conjecture of Birch and Swinnerton-Dyer), but these oscillations do not appear to have been noticed previously. This may be due in part to the critical role played by the conductor; in arithmetic statistics it is common to order elliptic curves E/Q by naive height rather than conductor, but doing so obscures these oscillations.
I will present results from an ongoing investigation of this phenomenon, which is remarkably robust and not specific to elliptic curves. One finds similar oscillations in the averages of Dirichlet coefficients of many types of L-functions when organized by conductor and root number, including those associated to modular forms and abelian varieties. The source of these murmurations in the case of weight-2 newforms with trivial nebentypus is now understood, thanks to recent work of Zubrilina, but all other cases remain open.
I will present results from an ongoing investigation of this phenomenon, which is remarkably robust and not specific to elliptic curves. One finds similar oscillations in the averages of Dirichlet coefficients of many types of L-functions when organized by conductor and root number, including those associated to modular forms and abelian varieties. The source of these murmurations in the case of weight-2 newforms with trivial nebentypus is now understood, thanks to recent work of Zubrilina, but all other cases remain open.
This is based on joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.
This is based on joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.
Link to recording (YouTube)
Link to slides
Alexander Mangerel, Correlations, sign patterns and rigidity theorems for multiplicative functions
(Durham University)
Alexander Mangerel, Correlations, sign patterns and rigidity theorems for multiplicative functions
(Durham University)
Thursday, September 7, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 8, 2023 (1am AEST, 3am NZST)
Thursday, September 7, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 8, 2023 (1am AEST, 3am NZST)
Abstract:
Abstract:
The Liouville function \lambda(n), defined to be +1 for n having an even number of prime factors (counted with multiplicity) and -1 otherwise, is a multiplicative function with deep connections to the distribution of primes. Inspired by the prime k-tuples conjecture of Hardy and Littlewood, Chowla conjectured that for every k each of the 2^k distinct sign patterns, i.e., tuples in \{-1,+1\}^k are assumed by the tuples (\lambda(n+1),...,\lambda(n+k)), n \in \mathbb{N}, with the same asymptotic frequency.
The Liouville function \lambda(n), defined to be +1 for n having an even number of prime factors (counted with multiplicity) and -1 otherwise, is a multiplicative function with deep connections to the distribution of primes. Inspired by the prime k-tuples conjecture of Hardy and Littlewood, Chowla conjectured that for every k each of the 2^k distinct sign patterns, i.e., tuples in \{-1,+1\}^k are assumed by the tuples (\lambda(n+1),...,\lambda(n+k)), n \in \mathbb{N}, with the same asymptotic frequency.
The underlying phenomenon at hand is that the prime factorisations of n+1,\ldots,n+k are expected to be (in a precise sense) statistically independent as n varies. As conjectured by Elliott, the same equidistribution of sign patterns is expected to hold for other \pm 1-valued multiplicative functions, provided they are ``far from being periodic''. To the best of our knowledge, until recently no explicit constructions of multiplicative functions with this behaviour were known.
The underlying phenomenon at hand is that the prime factorisations of n+1,\ldots,n+k are expected to be (in a precise sense) statistically independent as n varies. As conjectured by Elliott, the same equidistribution of sign patterns is expected to hold for other \pm 1-valued multiplicative functions, provided they are ``far from being periodic''. To the best of our knowledge, until recently no explicit constructions of multiplicative functions with this behaviour were known.
In this talk we will discuss precisely what Chowla's and Elliott's conjectures say, survey some of the literature on correlations, and discuss some related problems about sign patterns. Specifically, we will address:
In this talk we will discuss precisely what Chowla's and Elliott's conjectures say, survey some of the literature on correlations, and discuss some related problems about sign patterns. Specifically, we will address:
i) the construction of ``Liouville-like'' functions f: \mathbb{N} \rightarrow \{-1,+1\} whose k-tuples (f(n+1),...,f(n+k)) equidistribute in \{-1,+1\}^k, answering a question of de la Rue from 2018, and
i) the construction of ``Liouville-like'' functions f: \mathbb{N} \rightarrow \{-1,+1\} whose k-tuples (f(n+1),...,f(n+k)) equidistribute in \{-1,+1\}^k, answering a question of de la Rue from 2018, and
ii) in the case k = 4, the classification of all \pm 1-valued completely multiplicative functions f with the (rigid) property that the sequence of tuples (f(n+1),f(n+2),f(n+3),f(n+4)) omits the pattern (+1,+1,+1,+1), solving a 50-year old problem of R.H. Hudson.
ii) in the case k = 4, the classification of all \pm 1-valued completely multiplicative functions f with the (rigid) property that the sequence of tuples (f(n+1),f(n+2),f(n+3),f(n+4)) omits the pattern (+1,+1,+1,+1), solving a 50-year old problem of R.H. Hudson.
Key to these developments is a new result about the vanishing of correlations of ``moderately aperiodic'' multiplicative functions along a dense sequence of scales.
Key to these developments is a new result about the vanishing of correlations of ``moderately aperiodic'' multiplicative functions along a dense sequence of scales.
Based on joint work with O. Klurman and J. Teräväinen.
Based on joint work with O. Klurman and J. Teräväinen.
Link to recording (YouTube)
Link to slides
Alex Iosevich, Some number theoretic aspects of finite point configurations
(University of Rochester)
Alex Iosevich, Some number theoretic aspects of finite point configurations
(University of Rochester)
Thursday, June 29, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 30, 2023 (1am AEST, 3am NZST)
Thursday, June 29, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 30, 2023 (1am AEST, 3am NZST)
Abstract: We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centered around the Erdos/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area.
Abstract: We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centered around the Erdos/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area.
Link to slides
Εfthymios Sofos, The second moment method for rational points
(University of Glasgow)
Εfthymios Sofos, The second moment method for rational points
(University of Glasgow)
Thursday, June 22, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 23, 2023 (1am AEST, 3am NZST)
Thursday, June 22, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 23, 2023 (1am AEST, 3am NZST)
Abstract: In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen [arXiv:2212.10373] that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer--Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.
Abstract: In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen [arXiv:2212.10373] that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer--Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.
Link to slides
Youness Lamzouri, A walk on Legendre paths
(Institut Elie Cartan de Lorraine)
Youness Lamzouri, A walk on Legendre paths
(Institut Elie Cartan de Lorraine)
Thursday, June 15, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 16, 2023 (1am AEST, 3am NZST)
Thursday, June 15, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 16, 2023 (1am AEST, 3am NZST)
Abstract: In this talk, we shall explore certain polygonal paths, that we call ''Legendre paths'', which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, S_p(j)) for 0≤j≤p-1, where S_p(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis? We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions.
Abstract: In this talk, we shall explore certain polygonal paths, that we call ''Legendre paths'', which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, S_p(j)) for 0≤j≤p-1, where S_p(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis? We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions.
Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch.
Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch.
Carlo Pagano, On Chowla's non-vanishing conjecture over function fields
(Concordia University)
Carlo Pagano, On Chowla's non-vanishing conjecture over function fields
(Concordia University)
Thursday, June 8, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 9, 2023 (1am AEST, 3am NZST)
Thursday, June 8, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 9, 2023 (1am AEST, 3am NZST)
Abstract: A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz--Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui--Florea, David--Florea--Lalin, Ellenberg--Li--Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.
Abstract: A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz--Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui--Florea, David--Florea--Lalin, Ellenberg--Li--Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.
Lior Bary-Soroker, Random additive polynomials
(Tel Aviv University)
Lior Bary-Soroker, Random additive polynomials
(Tel Aviv University)
Thursday, June 1, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 2, 2023 (1am AEST, 3am NZST)
Thursday, June 1, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 2, 2023 (1am AEST, 3am NZST)
Abstract: Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e. the degree of the polynomial) goes to infinity.
Abstract: Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e. the degree of the polynomial) goes to infinity.
In this talk, we will discuss an important class of random polynomials — additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn.
In this talk, we will discuss an important class of random polynomials — additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn.
While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity.
While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity.
The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory.
The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory.
Based on a joint work with Alexei Entin and Eilidh McKemmie
Based on a joint work with Alexei Entin and Eilidh McKemmie
Barak Weiss, New bounds on lattice covering volumes, and nearly uniform covers
(Tel Aviv University)
Barak Weiss, New bounds on lattice covering volumes, and nearly uniform covers
(Tel Aviv University)
Thursday, May 25, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 26, 2023 (1am AEST, 3am NZST)
Thursday, May 25, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 26, 2023 (1am AEST, 3am NZST)
Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called “Geometry of Numbers”, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.
Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called “Geometry of Numbers”, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.
of the stated result.
of the stated result.
Mark Shusterman, Counting Minimally Ramified Global Field Extensions
(Harvard University)
Mark Shusterman, Counting Minimally Ramified Global Field Extensions
(Harvard University)
Thursday, May 18, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 19, 2023 (1am AEST, 3am NZST)
Thursday, May 18, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 19, 2023 (1am AEST, 3am NZST)
Abstract: Given a finite group G, one is interested in the number of Galois extensions of a global field with Galois group G and bounded discriminant. We consider a refinement of this problem where the discriminant is required to have the smallest possible number of (distinct) prime factors. We will discuss existing results and conjectures over number fields, and present some recent results over function fields.
Abstract: Given a finite group G, one is interested in the number of Galois extensions of a global field with Galois group G and bounded discriminant. We consider a refinement of this problem where the discriminant is required to have the smallest possible number of (distinct) prime factors. We will discuss existing results and conjectures over number fields, and present some recent results over function fields.
Link to slides
Ben Green, On Sarkozy's theorem for shifted primes
(University of Oxford)
Ben Green, On Sarkozy's theorem for shifted primes
(University of Oxford)
Thursday, May 11, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 12, 2023 (1am AEST, 3am NZST)
Thursday, May 11, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 12, 2023 (1am AEST, 3am NZST)
Abstract: Suppose that N is large and that A is a subset of {1,..,N} which does not contain two elements x, y with x - y equal to p-1, p a prime. Then A has cardinality at most N^{1 - c}, for some absolute c > 0. I will discuss the history of this kind of question as well as some aspects of the proof of the stated result.
Abstract: Suppose that N is large and that A is a subset of {1,..,N} which does not contain two elements x, y with x - y equal to p-1, p a prime. Then A has cardinality at most N^{1 - c}, for some absolute c > 0. I will discuss the history of this kind of question as well as some aspects of the proof of the stated result.
Peter Koymans, Counting nilpotent extensions
(University of Michigan)
Peter Koymans, Counting nilpotent extensions
(University of Michigan)
Thursday, May 4, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 5, 2023 (1am AEST, 3am NZST)
Thursday, May 4, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 5, 2023 (1am AEST, 3am NZST)
Abstract: We discuss some recent progress towards the strong form of Malle’s conjecture. Even for nilpotent extensions, only very few cases of this conjecture are currently known. We show how equidistribution of Frobenius elements plays an essential role in this problem and how this can be used to make further progress towards Malle’s conjecture. We will also discuss applications to the Massey vanishing conjecture and to lifting problems. This is joint work with Carlo Pagano.
Abstract: We discuss some recent progress towards the strong form of Malle’s conjecture. Even for nilpotent extensions, only very few cases of this conjecture are currently known. We show how equidistribution of Frobenius elements plays an essential role in this problem and how this can be used to make further progress towards Malle’s conjecture. We will also discuss applications to the Massey vanishing conjecture and to lifting problems. This is joint work with Carlo Pagano.
Pierre Le Boudec, 2-torsion in class groups of number fields
(University of Basel)
Pierre Le Boudec, 2-torsion in class groups of number fields
(University of Basel)
Thursday, April 27, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 28, 2023 (1am AEST, 3am NZST)
Thursday, April 27, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 28, 2023 (1am AEST, 3am NZST)
Abstract: It is well-known that the class number of a number field K of fixed degree n is roughly bounded by the square root of the absolute value of the discriminant of K. However, given a prime number p, the cardinality of the p-torsion subgroup of the class group of K is expected to be much smaller. Unfortunately, beating the trivial bound mentioned above is a hard problem. Indeed, this task had only been achieved for a handful of pairs (n,p) until Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao managed to do so for any degree n in the case p=2. In this talk we will go through their proof and we will present new bounds which depend on the geometry of the lattice underlying the ring of integers of K. This is joint work with Dante Bonolis.
Abstract: It is well-known that the class number of a number field K of fixed degree n is roughly bounded by the square root of the absolute value of the discriminant of K. However, given a prime number p, the cardinality of the p-torsion subgroup of the class group of K is expected to be much smaller. Unfortunately, beating the trivial bound mentioned above is a hard problem. Indeed, this task had only been achieved for a handful of pairs (n,p) until Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao managed to do so for any degree n in the case p=2. In this talk we will go through their proof and we will present new bounds which depend on the geometry of the lattice underlying the ring of integers of K. This is joint work with Dante Bonolis.
Bjorn Poonen, Integral points on curves via Baker's method and finite étale covers
(MIT)
Bjorn Poonen, Integral points on curves via Baker's method and finite étale covers
(MIT)
Thursday, April 20, 2023 (6am PDT, 9am EDT, 2pm GMT, 3pm CEST, 4pm Israel Daylight Time, 6:30pm Indian Standard Time, 9pm CST, 11pm AEST)
Friday, April 21, 2023 (1am NZST)
Thursday, April 20, 2023 (6am PDT, 9am EDT, 2pm GMT, 3pm CEST, 4pm Israel Daylight Time, 6:30pm Indian Standard Time, 9pm CST, 11pm AEST)
Friday, April 21, 2023 (1am NZST)
Abstract: We prove results in the direction of showing that for some affine curves, Baker's method applied to finite étale covers is insufficient to determine the integral points.
Abstract: We prove results in the direction of showing that for some affine curves, Baker's method applied to finite étale covers is insufficient to determine the integral points.
Hélène Esnault, Integrality Properties of the Betti Moduli Space
(Freie Universität Berlin)
Hélène Esnault, Integrality Properties of the Betti Moduli Space
(Freie Universität Berlin)
Thursday, April 13, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 14, 2023 (1am AEST, 3am NZST)
Thursday, April 13, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 14, 2023 (1am AEST, 3am NZST)
Abstract: We use de Jong’s conjecture and the existence of $\ell$-adic companions to single out integrality properties of the Betti moduli space. The first such instance was in joint work with Michael Groechenig on Simpson’s integrality conjecture for (cohomologically) rigid local systems. This integrality property yields an obstruction for a finitely presented group to be the fundamental group of a sooth quasi-projective complex variety. (joint with Johan de Jong)
Abstract: We use de Jong’s conjecture and the existence of $\ell$-adic companions to single out integrality properties of the Betti moduli space. The first such instance was in joint work with Michael Groechenig on Simpson’s integrality conjecture for (cohomologically) rigid local systems. This integrality property yields an obstruction for a finitely presented group to be the fundamental group of a sooth quasi-projective complex variety. (joint with Johan de Jong)
Kannan Soundararajan, Covering integers using quadratic forms
(Stanford University)
Kannan Soundararajan, Covering integers using quadratic forms
(Stanford University)
Thursday, April 6, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 7, 2023 (1am AEST, 3am NZST)
Thursday, April 6, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 7, 2023 (1am AEST, 3am NZST)
Abstract: How large must $\Delta$ be so that we can cover a substantial proportion of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ with $d$ below $\Delta$? Problems involving representations by binary quadratic forms have a long history, going back to Fermat. The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao. In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition: If $\Delta$ is below $(\log X)^{\log 2-\epsilon}$ then zero percent of the integers below $X$ are represented, whereas if $\Delta$ is above $(\log X)^{\log 2 +\epsilon}$ then 100 percent of the integers below $X$ are represented.
Abstract: How large must $\Delta$ be so that we can cover a substantial proportion of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ with $d$ below $\Delta$? Problems involving representations by binary quadratic forms have a long history, going back to Fermat. The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao. In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition: If $\Delta$ is below $(\log X)^{\log 2-\epsilon}$ then zero percent of the integers below $X$ are represented, whereas if $\Delta$ is above $(\log X)^{\log 2 +\epsilon}$ then 100 percent of the integers below $X$ are represented.
Ziyang Gao, Sparsity of rational and algebraic points
(Leibniz University Hannover)
Ziyang Gao, Sparsity of rational and algebraic points
(Leibniz University Hannover)
Thursday, March 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, March 31, 2023 (2am AEDT, 4am NZDT)
Thursday, March 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, March 31, 2023 (2am AEDT, 4am NZDT)
Abstract: It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.
Abstract: It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.
In my talk, I will explain the historical and recent developments of this problem according to the different grades.
In my talk, I will explain the historical and recent developments of this problem according to the different grades.
Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.
Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.
If time permits, I will mention the corresponding result in high dimensions.
If time permits, I will mention the corresponding result in high dimensions.
Ofir Gorodetsky, How many smooth numbers and smooth polynomials are there?
(University of Oxford)
Ofir Gorodetsky, How many smooth numbers and smooth polynomials are there?
(University of Oxford)
Thursday, March 23, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 23, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields.
Abstract: Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields.
We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)^2 (as conjectured by Hildebrand) and y=(log x)^(3/2), when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.
We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)^2 (as conjectured by Hildebrand) and y=(log x)^(3/2), when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.
Florian Luca, Recent progress on the Skolem problem
(University of the Witwatersrand)
Florian Luca, Recent progress on the Skolem problem
(University of the Witwatersrand)
Thursday, March 16, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 16, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. One of the main contributions of the talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method.
Abstract: The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. One of the main contributions of the talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method.
In the second part of the talk, we present the notion of an Universal Skolem Set, which is a subset of the positive integers on which the Skolem is decidable regardless of the linear recurrence. We give two examples of such sets, one of which is of positive density (that is, contains a positive proportion of all the positive integers).
In the second part of the talk, we present the notion of an Universal Skolem Set, which is a subset of the positive integers on which the Skolem is decidable regardless of the linear recurrence. We give two examples of such sets, one of which is of positive density (that is, contains a positive proportion of all the positive integers).
Alex Wilkie, Integer points on analytic sets
(University of Oxford and University of Manchester)
Alex Wilkie, Integer points on analytic sets
(University of Oxford and University of Manchester)
Thursday, March 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In 2004 I proved an O(loglogH) bound for the number of integer points of height at most H lying on a globally subanalytic curve. (The paper was published in the Journal of Symbolic Logic and so probably escaped the notice of most of you reading this.) Recently, Gareth Jones and Gal Binyamini proposed a generalization of the result to higher dimensions (where the obvious statement is almost certainly false) and I shall report on our joint work: one obtains the (hoped for) (loglogH)^n bound for (not globally subanalytic but) globally analytic sets of dimension n.
Abstract: In 2004 I proved an O(loglogH) bound for the number of integer points of height at most H lying on a globally subanalytic curve. (The paper was published in the Journal of Symbolic Logic and so probably escaped the notice of most of you reading this.) Recently, Gareth Jones and Gal Binyamini proposed a generalization of the result to higher dimensions (where the obvious statement is almost certainly false) and I shall report on our joint work: one obtains the (hoped for) (loglogH)^n bound for (not globally subanalytic but) globally analytic sets of dimension n.
Alexandra Shlapentokh, Defining integers using unit groups
(East Carolina University)
Alexandra Shlapentokh, Defining integers using unit groups
(East Carolina University)
Thursday, March 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 3, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 3, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We discuss some problems of definability and decidability over rings of integers of algebraic extensions of $\Q$. In particular, we show that for a large class of fields $K$ there is a simple formula defining rational integers over $O_K$. Below $U_K$ is the group of units of $O_K$.
Abstract: We discuss some problems of definability and decidability over rings of integers of algebraic extensions of $\Q$. In particular, we show that for a large class of fields $K$ there is a simple formula defining rational integers over $O_K$. Below $U_K$ is the group of units of $O_K$.
$\Z=\{x| \forall \varepsilon \in U_K\setminus \{1\}\ \exists \delta \in U_K: x \equiv \frac{\delta-1}{\varepsilon-1} \bmod (\varepsilon-1)\}$. This talk is based on a joint paper with Barry Mazur and Karl Rubin.
$\Z=\{x| \forall \varepsilon \in U_K\setminus \{1\}\ \exists \delta \in U_K: x \equiv \frac{\delta-1}{\varepsilon-1} \bmod (\varepsilon-1)\}$. This talk is based on a joint paper with Barry Mazur and Karl Rubin.
Terence Tao, Infinite Partial Sumsets in the Primes
(UCLA)
Terence Tao, Infinite Partial Sumsets in the Primes
(UCLA)
Thursday, February 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 24, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture). Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 < a_2 < ... and b_1 < b_2 < ... such that a_i + b_j is prime for all i<j. Equivalently, the primes are not "translation-finite" in the sense of Ruppert. As an application of these methods we show that the orbit closure of the primes is uncountable.
Abstract: It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture). Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 < a_2 < ... and b_1 < b_2 < ... such that a_i + b_j is prime for all i<j. Equivalently, the primes are not "translation-finite" in the sense of Ruppert. As an application of these methods we show that the orbit closure of the primes is uncountable.
Wanlin Li, Ordinary and Basic Reductions of Abelian Varieties
(Université de Montréal)
Wanlin Li, Ordinary and Basic Reductions of Abelian Varieties
(Université de Montréal)
Thursday, February 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc).
Abstract: Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc).
In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.
In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.
Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.
Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.
Sarah Peluse, Divisibility of character values of the symmetric group
(Institute for Advanced Study and Princeton University)
Sarah Peluse, Divisibility of character values of the symmetric group
(Institute for Advanced Study and Princeton University)
Thursday, February 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 10, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In 2017, Miller computed the character tables of $S_n$ for all $n$ up to $38$ and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of $S_n$ are divisible by any fixed prime power as $n$ tends to infinity. In this talk, I will discuss joint work with K. Soundararajan that resolves this conjecture, and mention some related open problems.
Abstract: In 2017, Miller computed the character tables of $S_n$ for all $n$ up to $38$ and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of $S_n$ are divisible by any fixed prime power as $n$ tends to infinity. In this talk, I will discuss joint work with K. Soundararajan that resolves this conjecture, and mention some related open problems.
Matilde Lalín, Distributions of sums of the divisor function over function fields
(Université de Montréal)
Matilde Lalín, Distributions of sums of the divisor function over function fields
(Université de Montréal)
Thursday, February 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 3, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 3, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick studied the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$. By results from the Katz and Sarnak philosophy, they were able to relate these problems to certain integrals over the ensemble of unitary matrices when $q$ goes to infinity. We study similar problems leading to integrals over the ensembles of symplectic and orthogonal matrices when $q$ goes to infinity. This is joint work with Vivian Kuperberg.
Abstract: In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick studied the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$. By results from the Katz and Sarnak philosophy, they were able to relate these problems to certain integrals over the ensemble of unitary matrices when $q$ goes to infinity. We study similar problems leading to integrals over the ensembles of symplectic and orthogonal matrices when $q$ goes to infinity. This is joint work with Vivian Kuperberg.
Robert Wilms, On equidistribution in Arakelov theory
(University of Basel)
Robert Wilms, On equidistribution in Arakelov theory
(University of Basel)
Thursday, January 26, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 27, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 26, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 27, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: As a motivating example of its own interest I will first discuss a new equidistribution result for the zero sets of integer polynomials. More precisely, I will give a condition such that the zero sets tends to equidistribute with respect to the Fubini-Study measure and I will show that this condition is generically satisfied in sets of polynomials of bounded Bombieri norm. In the second part, I will embed this example in a much more general framework about the distribution of the divisors of small sections of arithmetically ample hermitian line bundles in Arakelov theory.
Abstract: As a motivating example of its own interest I will first discuss a new equidistribution result for the zero sets of integer polynomials. More precisely, I will give a condition such that the zero sets tends to equidistribute with respect to the Fubini-Study measure and I will show that this condition is generically satisfied in sets of polynomials of bounded Bombieri norm. In the second part, I will embed this example in a much more general framework about the distribution of the divisors of small sections of arithmetically ample hermitian line bundles in Arakelov theory.
Cécile Dartyge, On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases
(Institut Élie Cartan, Université de Lorraine)
Cécile Dartyge, On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases
(Institut Élie Cartan, Université de Lorraine)
Thursday, January 19, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 20, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 19, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 20, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Let P(X) be a monic, quartic, irreducible polynomial of Z[X] with cyclic or dihedral Galois group. We prove that there exists c_P >0, such that for a positive proportion of integers n, P(n) has a prime factor bigger than n^{1+c_P}.
Abstract: Let P(X) be a monic, quartic, irreducible polynomial of Z[X] with cyclic or dihedral Galois group. We prove that there exists c_P >0, such that for a positive proportion of integers n, P(n) has a prime factor bigger than n^{1+c_P}.
This is a joint work with James Maynard.
This is a joint work with James Maynard.
Régis de la Bretèche, Higher moments of primes in arithmetic progressions
(Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Régis de la Bretèche, Higher moments of primes in arithmetic progressions
(Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Thursday, January 12, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 13, 2023 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 12, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 13, 2023 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In a joint work with Daniel Fiorilli, we develop a new method to prove lower bounds of some moments related to the distribution of primes in arithmetic progressions. We shall present main results and explain some aspects of the proofs. To prove our results, we assume GRH but we succeed to avoid linearly independence on zeroes hypothesis.
Abstract: In a joint work with Daniel Fiorilli, we develop a new method to prove lower bounds of some moments related to the distribution of primes in arithmetic progressions. We shall present main results and explain some aspects of the proofs. To prove our results, we assume GRH but we succeed to avoid linearly independence on zeroes hypothesis.
Umberto Zannier, Bounded generation in linear groups and exponential parametrizations
(Scuola Normale Superiore Pisa)
Umberto Zannier, Bounded generation in linear groups and exponential parametrizations
(Scuola Normale Superiore Pisa)
Thursday, December 22, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 23, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 22, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 23, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of “bounded generation” in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on “anisotropic groups” (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are “sparse”. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.
Abstract: In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of “bounded generation” in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on “anisotropic groups” (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are “sparse”. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.
William Duke, On the analytic theory of isotropic ternary quadratic forms
(UCLA)
William Duke, On the analytic theory of isotropic ternary quadratic forms
(UCLA)
Thursday, December 15, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 16, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 15, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 16, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is explicitly computed in terms of local data determined by the orbit. This is compared with the well-known asymptotic for the count of all primitive zeros. Together with an extension of results of Kneser by R. Schulze-Pillot on the classes in a genus of representations, this yields a formula for the number of orbits, summed over a genus of forms, in terms of the number of local orbits. For a certain special class of forms a simple explicit formula is given for this number.
Abstract: I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is explicitly computed in terms of local data determined by the orbit. This is compared with the well-known asymptotic for the count of all primitive zeros. Together with an extension of results of Kneser by R. Schulze-Pillot on the classes in a genus of representations, this yields a formula for the number of orbits, summed over a genus of forms, in terms of the number of local orbits. For a certain special class of forms a simple explicit formula is given for this number.
Laura DeMarco, Lattès maps, bifurcations, and arithmetic
(Harvard University)
Laura DeMarco, Lattès maps, bifurcations, and arithmetic
(Harvard University)
Thursday, December 8, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 9, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 8, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 9, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In the field of holomorphic dynamics, we learn that the Lattès maps -- the rational functions on P^1 that are quotients of maps on elliptic curves -- are rather boring. We can understand their dynamics completely. But viewed arithmetically, there are still unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on P^1 and, in turn, new perspectives on the arithmetic side. The new material is a joint project with Myrto Mavraki.
Abstract: In the field of holomorphic dynamics, we learn that the Lattès maps -- the rational functions on P^1 that are quotients of maps on elliptic curves -- are rather boring. We can understand their dynamics completely. But viewed arithmetically, there are still unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on P^1 and, in turn, new perspectives on the arithmetic side. The new material is a joint project with Myrto Mavraki.
Robert J. Lemke Oliver, Uniform exponent bounds on the number of primitive extensions of number fields
(Tufts University)
Robert J. Lemke Oliver, Uniform exponent bounds on the number of primitive extensions of number fields
(Tufts University)
Thursday, December 1, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 2, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 1, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 2, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach.
Abstract: A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach.
Here, we make progress on this weaker conjecture (which we term the ``uniform exponent conjecture'') in two ways. First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions. These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank. This is forthcoming work that grew out of conversations with M. Bhargava.
Here, we make progress on this weaker conjecture (which we term the ``uniform exponent conjecture'') in two ways. First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions. These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank. This is forthcoming work that grew out of conversations with M. Bhargava.
Jared Duker Lichtman, A proof of the Erdős primitive set conjecture
(University of Oxford)
Jared Duker Lichtman, A proof of the Erdős primitive set conjecture
(University of Oxford)
Thursday, November 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In 1986 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.
Abstract: A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In 1986 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.
Trevor Wooley, Waring’s Problem
(Purdue University)
Trevor Wooley, Waring’s Problem
(Purdue University)
Thursday, November 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In 1770, E. Waring made an assertion these days interpreted as conjecturing that when $k$ is a natural number, all positive integers may be written as the sum of a number $g(k)$ of positive integral $k$-th powers, with $g(k)$ finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding $G(k)$, the least number $s$ having the property that all sufficiently large integers can be written as the sum of $s$ positive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange), $G(3)\le 7$ (Linnik), $G(4)=16$ (Davenport), and $G(5)\le 17$, $G(6)\le 24$, ..., $G(20)\le 142$ (Vaughan and Wooley). For large $k$ one has $G(k)\le k(\log k+\log \log k+2+o(1))$ (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.
Abstract: In 1770, E. Waring made an assertion these days interpreted as conjecturing that when $k$ is a natural number, all positive integers may be written as the sum of a number $g(k)$ of positive integral $k$-th powers, with $g(k)$ finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding $G(k)$, the least number $s$ having the property that all sufficiently large integers can be written as the sum of $s$ positive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange), $G(3)\le 7$ (Linnik), $G(4)=16$ (Davenport), and $G(5)\le 17$, $G(6)\le 24$, ..., $G(20)\le 142$ (Vaughan and Wooley). For large $k$ one has $G(k)\le k(\log k+\log \log k+2+o(1))$ (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.
Link to slides
Emanuel Carneiro, Hilbert spaces and low-lying zeros of L-functions
(ICTP)
Emanuel Carneiro, Hilbert spaces and low-lying zeros of L-functions
(ICTP)
Thursday, November 10, 2022 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 10, 2022 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In this talk I would like to present some ideas behind a general Hilbert space framework for solving certain optimization problems that arise when studying the distribution of the low-lying zeros of families of $L$-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.
Abstract: In this talk I would like to present some ideas behind a general Hilbert space framework for solving certain optimization problems that arise when studying the distribution of the low-lying zeros of families of $L$-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.
Shai Evra, Optimal strong approximation and the Sarnak-Xue density hypothesis
(Hebrew University of Jerusalem)
Shai Evra, Optimal strong approximation and the Sarnak-Xue density hypothesis
(Hebrew University of Jerusalem)
Thursday, November 3, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 3, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: It is a classical result that the modulo map from SL_2(Z) to SL_2(Z/qZ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL_2(Z/qZ) has a lift in SL_2(Z) with coefficients of size at most q^e? A simple pigeonhole principle shows that e > 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL_2(Z). The proof relies on a density theorem of the Ramanujan conjecture for SL_2(Z).
Abstract: It is a classical result that the modulo map from SL_2(Z) to SL_2(Z/qZ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL_2(Z/qZ) has a lift in SL_2(Z) with coefficients of size at most q^e? A simple pigeonhole principle shows that e > 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL_2(Z). The proof relies on a density theorem of the Ramanujan conjecture for SL_2(Z).
In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands program.
In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands program.
This talk is based on ongoing joint works with B. Feigon, M. Gerbelli-Gauthier, H. Gustafssun, K. Maurischat and O. Parzanchevski.
This talk is based on ongoing joint works with B. Feigon, M. Gerbelli-Gauthier, H. Gustafssun, K. Maurischat and O. Parzanchevski.
Evelina Viada, Rational points on curves in a product of elliptic curves
(University of Göttingen)
Evelina Viada, Rational points on curves in a product of elliptic curves
(University of Göttingen)
Thursday, October 27, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 28, 2022 (2am AEDT, 4am NZST)
Thursday, October 27, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 28, 2022 (2am AEDT, 4am NZST)
Abstract: The Mordell-Conjecture (Faltings Theorem) states that an algebraic curve of genus at least 2 has only finitely many rational points. The Torsion Anomalous Conjecture (TAC) generalises Faltings Theorem. In some cases the proofs of the TAC are effective, implying effective cases of the Mordell-Conjecture. I would like to explain an effective method to determine the K-rational points on certain families of curves and to present some new specific examples. I will give an overview of the methods used in the context of the TAC presenting some general theorems and applications.
Abstract: The Mordell-Conjecture (Faltings Theorem) states that an algebraic curve of genus at least 2 has only finitely many rational points. The Torsion Anomalous Conjecture (TAC) generalises Faltings Theorem. In some cases the proofs of the TAC are effective, implying effective cases of the Mordell-Conjecture. I would like to explain an effective method to determine the K-rational points on certain families of curves and to present some new specific examples. I will give an overview of the methods used in the context of the TAC presenting some general theorems and applications.
Jack Thorne, Symmetric power functoriality for GL(2)
(University of Cambridge)
Jack Thorne, Symmetric power functoriality for GL(2)
(University of Cambridge)
Thursday, October 20, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 21, 2022 (2am AEDT, 4am NZST)
Thursday, October 20, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 21, 2022 (2am AEDT, 4am NZST)
Abstract: Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2) – the associated symmetric power L-functions are then the ones identified by Serre in the 1960’s in relation to the Sato—Tate conjecture.
Abstract: Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2) – the associated symmetric power L-functions are then the ones identified by Serre in the 1960’s in relation to the Sato—Tate conjecture.
I will describe the background to these ideas and then discuss the proof, joint with James Newton, of the existence of these symmetric power liftings for Hilbert modular forms. One arithmetic consequence is that if E is a (non-CM) elliptic curve over a real quadratic field, then all of its symmetric power L-functions admit analytic continuation to the whole complex plane.
I will describe the background to these ideas and then discuss the proof, joint with James Newton, of the existence of these symmetric power liftings for Hilbert modular forms. One arithmetic consequence is that if E is a (non-CM) elliptic curve over a real quadratic field, then all of its symmetric power L-functions admit analytic continuation to the whole complex plane.
Thomas Gauthier, A complex analytic approach to sparsity, rigidity and uniformity in arithmetic dynamics
(Université Paris-Saclay)
Thomas Gauthier, A complex analytic approach to sparsity, rigidity and uniformity in arithmetic dynamics
(Université Paris-Saclay)
Thursday, October 13, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 14, 2022 (2am AEDT, 4am NZST)
Thursday, October 13, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 14, 2022 (2am AEDT, 4am NZST)
Abstract: This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, I will give a strategy to tackle several of those problems at the same time. This is based on a joint work in progress with Gabriel Vigny and Johan Taflin.
Abstract: This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, I will give a strategy to tackle several of those problems at the same time. This is based on a joint work in progress with Gabriel Vigny and Johan Taflin.
Jeffrey C. Lagarias, The Alternative Hypothesis and Point Processes
(University of Michigan)
Jeffrey C. Lagarias, The Alternative Hypothesis and Point Processes
(University of Michigan)
Thursday, October 6, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 7, 2022 (2am AEDT, 4am NZST)
Thursday, October 6, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 7, 2022 (2am AEDT, 4am NZST)
Abstract: The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that nearly all normalized zero spacings be near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to the long-standing problem of proving there are no exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing Alternative Hypothesis type statistics, which is consistent with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by Tao with slightly different methods.) The talk reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.
Abstract: The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that nearly all normalized zero spacings be near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to the long-standing problem of proving there are no exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing Alternative Hypothesis type statistics, which is consistent with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by Tao with slightly different methods.) The talk reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.
Paul Nelson, The sup norm problem in the level aspect
(Aarhus University)
Paul Nelson, The sup norm problem in the level aspect
(Aarhus University)
Thursday, September 29, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 30, 2022 (1am AEST, 4am NZDT)
Thursday, September 29, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 30, 2022 (1am AEST, 4am NZDT)
Abstract: The sup norm problem concerns the size of L^2-normalized eigenfunctions of manifolds. In many situations, one expects to be able to improve upon the general bound following from local considerations. The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface. Their method has since been extended and applied by many authors, notably to the "level aspect" variant of the problem, where one varies the underlying manifold rather than the eigenvalue. Recently, Raphael Steiner introduced a new method for attacking the sup norm problem. I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.
Abstract: The sup norm problem concerns the size of L^2-normalized eigenfunctions of manifolds. In many situations, one expects to be able to improve upon the general bound following from local considerations. The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface. Their method has since been extended and applied by many authors, notably to the "level aspect" variant of the problem, where one varies the underlying manifold rather than the eigenvalue. Recently, Raphael Steiner introduced a new method for attacking the sup norm problem. I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.
Alexandra Florea, Negative moments of the Riemann zeta function
(University of California Irvine)
Alexandra Florea, Negative moments of the Riemann zeta function
(University of California Irvine)
Thursday, September 22, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 23, 2022 (1am AEST, 3am NZST)
Thursday, September 22, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 23, 2022 (1am AEST, 3am NZST)
Abstract: I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.
Abstract: I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.
Danny Neftin, Reducible fibers of polynomial maps
(Technion-Israel Institute of Technology)
Danny Neftin, Reducible fibers of polynomial maps
(Technion-Israel Institute of Technology)
Thursday, September 15, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 16, 2022 (1am AEST, 3am NZST)
Thursday, September 15, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 16, 2022 (1am AEST, 3am NZST)
Abstract: For a polynomial $f\in \mathbb Q[x]$, the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a ``thin" set of exceptions $R_f$ whose explicit description is unknown in general. The problem of describing $R_f$ is closely related to reducibility and arboreal representations in arithmetic dynamics, as well as to Kronecker and arithmetic equivalence for polynomial maps, that is, polynomial versions of the question: "can you hear the shape of the drum?". We shall discuss recent progress on the above problem and topics.
Abstract: For a polynomial $f\in \mathbb Q[x]$, the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a ``thin" set of exceptions $R_f$ whose explicit description is unknown in general. The problem of describing $R_f$ is closely related to reducibility and arboreal representations in arithmetic dynamics, as well as to Kronecker and arithmetic equivalence for polynomial maps, that is, polynomial versions of the question: "can you hear the shape of the drum?". We shall discuss recent progress on the above problem and topics.
Ping Xi, Analytic approaches towards Katz’s problems on Kloosterman sums
(Xi'an Jiaotong University)
Ping Xi, Analytic approaches towards Katz’s problems on Kloosterman sums
(Xi'an Jiaotong University)
Thursday, September 8, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 9, 2022 (1am AEST, 3am NZST)
Thursday, September 8, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 9, 2022 (1am AEST, 3am NZST)
Abstract: Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from $\ell$-adic cohomology.
Abstract: Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from $\ell$-adic cohomology.
Yann Bugeaud, B'
(University of Strasbourg)
Yann Bugeaud, B'
(University of Strasbourg)
Thursday, September 1, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 2, 2022 (1am AEST, 3am NZST)
Thursday, September 1, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 2, 2022 (1am AEST, 3am NZST)
Alexander Gamburd, Arithmetic and dynamics on varieties of Markoff type
(CUNY Graduate Center)
Alexander Gamburd, Arithmetic and dynamics on varieties of Markoff type
(CUNY Graduate Center)
Thursday, June 30, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 1, 2022 (1am AEST, 3am NZST)
Thursday, June 30, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 1, 2022 (1am AEST, 3am NZST)
Abstract: The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.
Abstract: The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.
Amir Shpilka, Points, lines and polynomial identities
(Tel Aviv University)
Amir Shpilka, Points, lines and polynomial identities
(Tel Aviv University)
Thursday, June 23, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 24, 2022 (1am AEST, 3am NZST)
Thursday, June 23, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 24, 2022 (1am AEST, 3am NZST)
Abstract: The Sylvester-Gallai (SG) theorem in discrete geometry asserts that if a finite set of points P has the property that every line through any two of its points intersects the set at a third point, then P must lie on a line. Surprisingly, this theorem, and some variants of it, appear in the analysis of locally correctable codes and, more noticeably, in algebraic program testing (polynomial identity testing). For these questions one often has to study extensions of the original SG problem: the case where there are several sets, or with a robust version of the condition (many "special" lines through each point) or with a higher degree analog of the problem, etc.
Abstract: The Sylvester-Gallai (SG) theorem in discrete geometry asserts that if a finite set of points P has the property that every line through any two of its points intersects the set at a third point, then P must lie on a line. Surprisingly, this theorem, and some variants of it, appear in the analysis of locally correctable codes and, more noticeably, in algebraic program testing (polynomial identity testing). For these questions one often has to study extensions of the original SG problem: the case where there are several sets, or with a robust version of the condition (many "special" lines through each point) or with a higher degree analog of the problem, etc.
In this talk I will present the SG theorem and some of its variants, show its relation to the above mentioned computational problems and discuss recent developments regarding higher degree analogs and their applications.
In this talk I will present the SG theorem and some of its variants, show its relation to the above mentioned computational problems and discuss recent developments regarding higher degree analogs and their applications.
John Voight, Counting elliptic curves with level structure
(Dartmouth College)
John Voight, Counting elliptic curves with level structure
(Dartmouth College)
Thursday, June 16, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 17, 2022 (1am AEST, 3am NZST)
Thursday, June 16, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 17, 2022 (1am AEST, 3am NZST)
Abstract: Following work of Harron and Snowden, we provide an asymptotic answer to questions like: how many elliptic curves of bounded height have a cyclic isogeny of degree N? We'll begin with a survey the recent spate of work on this topic, and then we will report on joint work with Carl Pomerance and Maggie Pizzo, with John Cullinan and Meagan Kenney, and finally with Grant Molnar.
Abstract: Following work of Harron and Snowden, we provide an asymptotic answer to questions like: how many elliptic curves of bounded height have a cyclic isogeny of degree N? We'll begin with a survey the recent spate of work on this topic, and then we will report on joint work with Carl Pomerance and Maggie Pizzo, with John Cullinan and Meagan Kenney, and finally with Grant Molnar.
Nicole Looper, The Uniform Boundedness Principle for polynomials over number fields
(Brown University)
Nicole Looper, The Uniform Boundedness Principle for polynomials over number fields
(Brown University)
Thursday, June 9, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 10, 2022 (1am AEST, 3am NZST)
Thursday, June 9, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 10, 2022 (1am AEST, 3am NZST)
Abstract: This talk is about uniform bounds on the number of K-rational preperiodic points across families of endomorphisms of projective space defined over various fields K. We will focus on the case where K is a number field, and the morphisms are polynomial maps on P^1. Along the way, I will highlight the more challenging aspects behind the known approaches, and discuss the obstacles to be addressed in future research.
Abstract: This talk is about uniform bounds on the number of K-rational preperiodic points across families of endomorphisms of projective space defined over various fields K. We will focus on the case where K is a number field, and the morphisms are polynomial maps on P^1. Along the way, I will highlight the more challenging aspects behind the known approaches, and discuss the obstacles to be addressed in future research.
Elon Lindenstrauss, Effective equidistribution of some unipotent flows with polynomial rates
(Hebrew University of Jerusalem)
Elon Lindenstrauss, Effective equidistribution of some unipotent flows with polynomial rates
(Hebrew University of Jerusalem)
Thursday, June 2, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 3, 2022 (1am AEST, 3am NZST)
Thursday, June 2, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 3, 2022 (1am AEST, 3am NZST)
Abstract: Joint work with Amir Mohammadi and Zhiren Wang
Abstract: Joint work with Amir Mohammadi and Zhiren Wang
A landmark result of Ratner gives that if $G$ is a real linear algebraic group, $\Gamma$ a lattice in $G$ and if $u_t$ is a one-parameter unipotent subgroup of $G$, then for any $x \in G/\Gamma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some subgroup $L <G$, and moreover that the orbit of $x$ under $u_t$ is contained in this periodic $L$ orbit.
A landmark result of Ratner gives that if $G$ is a real linear algebraic group, $\Gamma$ a lattice in $G$ and if $u_t$ is a one-parameter unipotent subgroup of $G$, then for any $x \in G/\Gamma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some subgroup $L <G$, and moreover that the orbit of $x$ under $u_t$ is contained in this periodic $L$ orbit.
A key motivation behind Ratner's equidistribution theorem for one-parameter unipotent flows has been to establish Raghunathan's conjecture regarding the possible orbit closures of groups generated by one-parameter unipotent groups; using the equidistribution theorem Ratner proved that if $G$ and $\Gamma$ are as above, and if $H<G$ is generated by one parameter unipotent groups then for any $x \in G/\Gamma$ one has that $\overline{H.x}=L.x$ where $H < L < G$ and $L.x$ is periodic. Important special cases of Raghunathan's conjecture were proven earlier by Margulis and by Dani and Margulis by a different, more direct, approach.
A key motivation behind Ratner's equidistribution theorem for one-parameter unipotent flows has been to establish Raghunathan's conjecture regarding the possible orbit closures of groups generated by one-parameter unipotent groups; using the equidistribution theorem Ratner proved that if $G$ and $\Gamma$ are as above, and if $H<G$ is generated by one parameter unipotent groups then for any $x \in G/\Gamma$ one has that $\overline{H.x}=L.x$ where $H < L < G$ and $L.x$ is periodic. Important special cases of Raghunathan's conjecture were proven earlier by Margulis and by Dani and Margulis by a different, more direct, approach.
These results have had many beautiful and unexpected applications in number theory, geometry and other areas. A key challenge has been to quantify and effectify these results. Beyond the case of actions of horospheric groups where there are several fully quantitative and effective results available, results in this direction have been few and far between. In particular, if $G$ is semisimple and $U$ is not horospheric no quantitative form of Ratner's equidistribution was known with any error rate, though there has been some progress on understanding quantitatively density properties of such flows with iterative logarithm error rates.
These results have had many beautiful and unexpected applications in number theory, geometry and other areas. A key challenge has been to quantify and effectify these results. Beyond the case of actions of horospheric groups where there are several fully quantitative and effective results available, results in this direction have been few and far between. In particular, if $G$ is semisimple and $U$ is not horospheric no quantitative form of Ratner's equidistribution was known with any error rate, though there has been some progress on understanding quantitatively density properties of such flows with iterative logarithm error rates.
In my talk I will present a new fully quantitative and effective equidistribution result for orbits of one-parameter unipotent groups in arithmetic quotients of $\SL_2(\C)$ and $\SL_2(\R)\times\SL(2,\R)$. I will also try to explain a bit the connection to number theory.
In my talk I will present a new fully quantitative and effective equidistribution result for orbits of one-parameter unipotent groups in arithmetic quotients of $\SL_2(\C)$ and $\SL_2(\R)\times\SL(2,\R)$. I will also try to explain a bit the connection to number theory.
Yunqing Tang, Applications of arithmetic holonomicity theorems
(Princeton University)
Yunqing Tang, Applications of arithmetic holonomicity theorems
(Princeton University)
Thursday, May 26, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 27, 2022 (1am AEST, 3am NZST)
Thursday, May 26, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 27, 2022 (1am AEST, 3am NZST)
Abstract: In this talk, we will discuss the proof of the unbounded denominators conjecture on Fourier coefficients of SL_2(Z)-modular forms, and the proof of irrationality of 2-adic zeta value at 5. Both proofs use an arithmetic holonomicity theorem, which can be viewed as a refinement of André’s algebraicity criterion. If time permits, we will give a proof of the arithmetic holonomicity theorem via the slope method a la Bost.
Abstract: In this talk, we will discuss the proof of the unbounded denominators conjecture on Fourier coefficients of SL_2(Z)-modular forms, and the proof of irrationality of 2-adic zeta value at 5. Both proofs use an arithmetic holonomicity theorem, which can be viewed as a refinement of André’s algebraicity criterion. If time permits, we will give a proof of the arithmetic holonomicity theorem via the slope method a la Bost.
This is joint work with Frank Calegari and Vesselin Dimitrov.
This is joint work with Frank Calegari and Vesselin Dimitrov.
Jeffrey Vaaler, Schinzel's determinant inequality and a conjecture of F. Rodriguez Villegas
(University of Texas at Austin)
Jeffrey Vaaler, Schinzel's determinant inequality and a conjecture of F. Rodriguez Villegas
(University of Texas at Austin)
Thursday, May 19, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 20, 2022 (1am AEST, 3am NZST)
Thursday, May 19, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 20, 2022 (1am AEST, 3am NZST)
Robert Charles Vaughan, Generalizations of the Montgomery-Hooley asymptotic formula; A survey.
(Pennsylvania State University)
Robert Charles Vaughan, Generalizations of the Montgomery-Hooley asymptotic formula; A survey.
(Pennsylvania State University)
Thursday, May 12, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 13, 2022 (1am AEST, 3am NZST)
Thursday, May 12, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 13, 2022 (1am AEST, 3am NZST)
Levent Alpöge, On integers which are(n't) the sum of two rational cubes
(Harvard University)
Levent Alpöge, On integers which are(n't) the sum of two rational cubes
(Harvard University)
Thursday, May 5, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 6, 2022 (1am AEST, 3am NZST)
Thursday, May 5, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 6, 2022 (1am AEST, 3am NZST)
Abstract: It's easy that 0% of integers are the sum of two integral cubes (allowing opposite signs!).
Abstract: It's easy that 0% of integers are the sum of two integral cubes (allowing opposite signs!).
I will explain joint work with Bhargava and Shnidman in which we show:
I will explain joint work with Bhargava and Shnidman in which we show:
1. At least a sixth of integers are not the sum of two rational cubes,
1. At least a sixth of integers are not the sum of two rational cubes,
and
and
2. At least a sixth of odd integers are the sum of two rational cubes!
2. At least a sixth of odd integers are the sum of two rational cubes!
(--- with 2. relying on new 2-converse results of Burungale-Skinner.)
(--- with 2. relying on new 2-converse results of Burungale-Skinner.)
The basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradict e.g. 1., and we show this by using the circle method "inside" the usual geometry of numbers argument applied to a particular coregular representation. Even then the resulting constant isn't small enough to conclude 1., so we use the clean form of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.
The basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradict e.g. 1., and we show this by using the circle method "inside" the usual geometry of numbers argument applied to a particular coregular representation. Even then the resulting constant isn't small enough to conclude 1., so we use the clean form of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.
Andrew Granville, Linear Divisibility sequences
(Université de Montréal)
Andrew Granville, Linear Divisibility sequences
(Université de Montréal)
Thursday, April 28, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 29, 2022 (1am AEST, 3am NZST)
Thursday, April 28, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 29, 2022 (1am AEST, 3am NZST)
Abstract: In 1878, in the first volume of the first mathematics journal published in the US, Edouard Lucas wrote 88 pages (in French) on linear recurrence sequences, placing Fibonacci numbers and other linear recurrence sequences into a broader context. He examined their behaviour locally as well as globally, and asked several questions that influenced much research in the century and a half to come.
Abstract: In 1878, in the first volume of the first mathematics journal published in the US, Edouard Lucas wrote 88 pages (in French) on linear recurrence sequences, placing Fibonacci numbers and other linear recurrence sequences into a broader context. He examined their behaviour locally as well as globally, and asked several questions that influenced much research in the century and a half to come.
In a sequence of papers in the 1930s, Marshall Hall further developed several of Lucas' themes, including studying and trying to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the additional property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable to even conjecture what a general theorem should look like, and despite developments over the years by various authors, such as Lehmer, Morgan Ward, van der Poorten, Bezivin, Petho, Richard Guy, Hugh Williams,... with higher order linear divisibility sequences, even the formulation of the classification has remained mysterious.
In a sequence of papers in the 1930s, Marshall Hall further developed several of Lucas' themes, including studying and trying to classify third order linear divisibility sequences; that is, linear recurrences like the Fibonacci numbers which have the additional property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases, Hall was unable to even conjecture what a general theorem should look like, and despite developments over the years by various authors, such as Lehmer, Morgan Ward, van der Poorten, Bezivin, Petho, Richard Guy, Hugh Williams,... with higher order linear divisibility sequences, even the formulation of the classification has remained mysterious.
In this talk we present our ongoing efforts to classify all linear divisibility sequences, the key new input coming from a wonderful application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation, due to Corvaja and Zannier.
In this talk we present our ongoing efforts to classify all linear divisibility sequences, the key new input coming from a wonderful application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation, due to Corvaja and Zannier.
Joni Teräväinen, Short exponential sums of the primes
(University of Turku)
Joni Teräväinen, Short exponential sums of the primes
(University of Turku)
Thursday, April 21, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 22, 2022 (1am AEST, 3am NZST)
Thursday, April 21, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 22, 2022 (1am AEST, 3am NZST)
Abstract: I will discuss the short interval behaviour of the von Mangoldt and Möbius functions twisted by exponentials. I will in particular mention new results on sums of these functions twisted by polynomial exponential phases, or even more general nilsequence phases. I will also discuss connections to Chowla's conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao, Terence Tao and Tamar Ziegler.
Abstract: I will discuss the short interval behaviour of the von Mangoldt and Möbius functions twisted by exponentials. I will in particular mention new results on sums of these functions twisted by polynomial exponential phases, or even more general nilsequence phases. I will also discuss connections to Chowla's conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao, Terence Tao and Tamar Ziegler.
Ram Murty, Probability Theory and the Riemann Hypothesis
(Queen's University)
Ram Murty, Probability Theory and the Riemann Hypothesis
(Queen's University)
Thursday, April 14, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 15, 2022 (1am AEST, 3am NZST)
Thursday, April 14, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 15, 2022 (1am AEST, 3am NZST)
Abstract: There is a probability distribution attached to the Riemann zeta function which allows one to formulate the Riemann hypothesis in terms of the cumulants of this distribution and is due to Biane, Pitman and Yor. The cumulants can be related to generalized Euler-Stieltjes constants and to Li's criterion for the Riemann hypothesis. We will discuss these results and present some new results related to this theme.
Abstract: There is a probability distribution attached to the Riemann zeta function which allows one to formulate the Riemann hypothesis in terms of the cumulants of this distribution and is due to Biane, Pitman and Yor. The cumulants can be related to generalized Euler-Stieltjes constants and to Li's criterion for the Riemann hypothesis. We will discuss these results and present some new results related to this theme.
Ana Caraiani, On the cohomology of Shimura varieties with torsion coefficients
(Imperial College London)
Ana Caraiani, On the cohomology of Shimura varieties with torsion coefficients
(Imperial College London)
Thursday, April 7, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 8, 2022 (1am AEST, 3am NZST)
Thursday, April 7, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 8, 2022 (1am AEST, 3am NZST)
Abstract: Shimura varieties are certain highly symmetric algebraic varieties that generalise modular curves and that play an important role in the Langlands program. In this talk, I will survey recent vanishing conjectures and results about the cohomology of Shimura varieties with torsion coefficients, under both local and global representation-theoretic conditions. I will illustrate the geometric ingredients needed to establish these results using the toy model of the modular curve. I will also mention several applications, including to (potential) modularity over CM fields.
Abstract: Shimura varieties are certain highly symmetric algebraic varieties that generalise modular curves and that play an important role in the Langlands program. In this talk, I will survey recent vanishing conjectures and results about the cohomology of Shimura varieties with torsion coefficients, under both local and global representation-theoretic conditions. I will illustrate the geometric ingredients needed to establish these results using the toy model of the modular curve. I will also mention several applications, including to (potential) modularity over CM fields.
William Chen, Markoff triples and connectivity of Hurwitz spaces
(Institute for Advanced Study)
William Chen, Markoff triples and connectivity of Hurwitz spaces
(Institute for Advanced Study)
Thursday, March 31, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 1, 2022 (2am AEDT, 4am NZDT)
Thursday, March 31, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 1, 2022 (2am AEDT, 4am NZDT)
Abstract: In this talk we will show that the integral points of the Markoff equation x^2 + y^2 + z^2 - xyz = 0 surject onto its F_p-points for all but finitely many primes p. This essentially resolves a conjecture of Bourgain, Gamburd, and Sarnak, and a question of Frobenius from 1913. The proof relates the question to the classical problem of classifying the connected components of the Hurwitz moduli spaces H(g,n) classifying finite covers of genus g curves with n branch points. Over a century ago, Clebsch and Hurwitz established connectivity for the subspace classifying simply branched covers of the projective line, which led to the first proof of the irreducibility of the moduli space of curves of a given genus. More recently, the work of Dunfield-Thurston and Conway-Parker establish connectivity in certain situations where the monodromy group is fixed and either g or n are allowed to be large, which has been applied to study Cohen-Lenstra heuristics over function fields. In the case where (g,n) are fixed and the monodromy group is allowed to vary, far less is known. In our case we study SL(2,p)-covers of elliptic curves, only branched over the origin, and establish connectivity, for all sufficiently large p, of the subspace classifying those covers with ramification indices 2p. The proof builds upon asymptotic results of Bourgain, Gamburd, and Sarnak, the key new ingredient being a divisibility result on the degree of a certain forgetful map between moduli spaces, which provides enough rigidity to bootstrap their asymptotics to a result for all sufficiently large p.
Abstract: In this talk we will show that the integral points of the Markoff equation x^2 + y^2 + z^2 - xyz = 0 surject onto its F_p-points for all but finitely many primes p. This essentially resolves a conjecture of Bourgain, Gamburd, and Sarnak, and a question of Frobenius from 1913. The proof relates the question to the classical problem of classifying the connected components of the Hurwitz moduli spaces H(g,n) classifying finite covers of genus g curves with n branch points. Over a century ago, Clebsch and Hurwitz established connectivity for the subspace classifying simply branched covers of the projective line, which led to the first proof of the irreducibility of the moduli space of curves of a given genus. More recently, the work of Dunfield-Thurston and Conway-Parker establish connectivity in certain situations where the monodromy group is fixed and either g or n are allowed to be large, which has been applied to study Cohen-Lenstra heuristics over function fields. In the case where (g,n) are fixed and the monodromy group is allowed to vary, far less is known. In our case we study SL(2,p)-covers of elliptic curves, only branched over the origin, and establish connectivity, for all sufficiently large p, of the subspace classifying those covers with ramification indices 2p. The proof builds upon asymptotic results of Bourgain, Gamburd, and Sarnak, the key new ingredient being a divisibility result on the degree of a certain forgetful map between moduli spaces, which provides enough rigidity to bootstrap their asymptotics to a result for all sufficiently large p.
Winnie Li, Group based zeta functions
(Pennsylvania State University)
Winnie Li, Group based zeta functions
(Pennsylvania State University)
Thursday, March 24, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 24, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The theme of this survey talk is zeta functions which count closed geodesics on objects arising from real and p-adic groups. Our focus is on PGL(n). For PGL(2), these are the Selberg zeta function for compact quotients of the upper half-plane and the Ihara zeta function for finite regular graphs. We shall explain the identities satisfied by these zeta functions, which show interconnections between combinatorics, group theory and number theory. Comparisons will be made for zeta identities from different background. Like the Riemann zeta function, the analytic behavior of a group based zeta function governs the distribution of the prime geodesics in its definition.
Abstract: The theme of this survey talk is zeta functions which count closed geodesics on objects arising from real and p-adic groups. Our focus is on PGL(n). For PGL(2), these are the Selberg zeta function for compact quotients of the upper half-plane and the Ihara zeta function for finite regular graphs. We shall explain the identities satisfied by these zeta functions, which show interconnections between combinatorics, group theory and number theory. Comparisons will be made for zeta identities from different background. Like the Riemann zeta function, the analytic behavior of a group based zeta function governs the distribution of the prime geodesics in its definition.
Aaron Levin, Diophantine Approximation for Closed Subschemes
(Michigan State University)
Aaron Levin, Diophantine Approximation for Closed Subschemes
(Michigan State University)
Thursday, March 17, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 17, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The classical Weil height machine associates heights to divisors on a projective variety. I will give a brief, but gentle, introduction to how this machinery extends to objects (closed subschemes) in higher codimension, due to Silverman, and discuss various ways to interpret the heights. We will then discuss several recent results in which these ideas play a prominent and central role.
Abstract: The classical Weil height machine associates heights to divisors on a projective variety. I will give a brief, but gentle, introduction to how this machinery extends to objects (closed subschemes) in higher codimension, due to Silverman, and discuss various ways to interpret the heights. We will then discuss several recent results in which these ideas play a prominent and central role.
Dmitry Kleinbock, Shrinking targets on homogeneous spaces and improving Dirichlet's Theorem
(Brandeis University)
Dmitry Kleinbock, Shrinking targets on homogeneous spaces and improving Dirichlet's Theorem
(Brandeis University)
Thursday, March 10, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 10, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Let $\psi$ be a decreasing function defined on all large positive real numbers. We say that a real $m \times n$ matrix $Y$ is "$\psi$-Dirichlet" if for every sufficiently large real number $T$ there exist non-trivial integer vectors $(p,q)$ satisfying $\|Yq-p\|^m < \psi(T)$ and $\|q\|^n < T$ (where $\|\cdot\|$ denotes the supremum norm on vectors). This generalizes the property of $Y$ being "Dirichlet improvable" which has been studied by several people, starting with Davenport and Schmidt in 1969. I will present results giving sufficient conditions on $\psi$ to ensure that the set of $\psi$-Dirichlet matrices has zero (resp., full) measure. If time allows I will mention a geometric generalization of the set-up, where the supremum norm is replaced by an arbitrary norm. Joint work with Anurag Rao, Andreas Strombergsson, Nick Wadleigh and Shuchweng Yu.
Abstract: Let $\psi$ be a decreasing function defined on all large positive real numbers. We say that a real $m \times n$ matrix $Y$ is "$\psi$-Dirichlet" if for every sufficiently large real number $T$ there exist non-trivial integer vectors $(p,q)$ satisfying $\|Yq-p\|^m < \psi(T)$ and $\|q\|^n < T$ (where $\|\cdot\|$ denotes the supremum norm on vectors). This generalizes the property of $Y$ being "Dirichlet improvable" which has been studied by several people, starting with Davenport and Schmidt in 1969. I will present results giving sufficient conditions on $\psi$ to ensure that the set of $\psi$-Dirichlet matrices has zero (resp., full) measure. If time allows I will mention a geometric generalization of the set-up, where the supremum norm is replaced by an arbitrary norm. Joint work with Anurag Rao, Andreas Strombergsson, Nick Wadleigh and Shuchweng Yu.
Ekin Özman, Modular Curves and Asymptotic Solutions to Fermat-type Equations
(Boğaziçi University)
Ekin Özman, Modular Curves and Asymptotic Solutions to Fermat-type Equations
(Boğaziçi University)
Thursday, March 3, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, March 3, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, March 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations, it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve play a central role in this approach. It is also possible to study the solutions of Fermat type equations over number fields asymptotically. In this talk, I will mention some recent results about these notions for the classical Fermat equation as well as some other Diophantine equations.
Abstract: Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations, it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve play a central role in this approach. It is also possible to study the solutions of Fermat type equations over number fields asymptotically. In this talk, I will mention some recent results about these notions for the classical Fermat equation as well as some other Diophantine equations.
Igor Shparlinski, Sums of Kloosterman and Salie Sums and Moments of L-functions
(UNSW Sydney)
Igor Shparlinski, Sums of Kloosterman and Salie Sums and Moments of L-functions
(UNSW Sydney)
Thursday, February 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 25, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We present some old and more recent results which suggest that Kloosterman and Salie sums exhibit a pseudorandom behaviour similar to the behaviour which is traditionally attributed to the Mobius function. In particular, we formulate some analogues of the Chowla Conjecture for Kloosterman and Salie sums. We then describe several results about the non-correlation of Kloosterman and Salie sums between themselves and also with some classical number-theoretic functions such as the Mobius function, the divisor function and the sums of binary digits. Various arithmetic applications of these results, including to asymptotic formulas for moments of various L-functions, will be outlined as well.
Abstract: We present some old and more recent results which suggest that Kloosterman and Salie sums exhibit a pseudorandom behaviour similar to the behaviour which is traditionally attributed to the Mobius function. In particular, we formulate some analogues of the Chowla Conjecture for Kloosterman and Salie sums. We then describe several results about the non-correlation of Kloosterman and Salie sums between themselves and also with some classical number-theoretic functions such as the Mobius function, the divisor function and the sums of binary digits. Various arithmetic applications of these results, including to asymptotic formulas for moments of various L-functions, will be outlined as well.
Harry Schmidt, Counting rational points and lower bounds for Galois orbits for special points on Shimura varieties
(University of Basel)
Harry Schmidt, Counting rational points and lower bounds for Galois orbits for special points on Shimura varieties
(University of Basel)
Thursday, February 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 18, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In this talk I will give an overview of the history of the André-Oort conjecture and its resolution last year after the final steps were made in work of Pila, Shankar, Tsimerman, Esnault and Groechenig as well as Binyamini, Yafaev and myself. I will focus on the key insights and ideas related to model theory and transcendence theory.
Abstract: In this talk I will give an overview of the history of the André-Oort conjecture and its resolution last year after the final steps were made in work of Pila, Shankar, Tsimerman, Esnault and Groechenig as well as Binyamini, Yafaev and myself. I will focus on the key insights and ideas related to model theory and transcendence theory.
Link to recording (112MB) including introductory words by Andrei Yafaev on the recent passing of Bas Edixhoven.
Link to recording (112MB) including introductory words by Andrei Yafaev on the recent passing of Bas Edixhoven.
Zeev Rudnick, Beyond uniform distribution
(Tel Aviv University)
Zeev Rudnick, Beyond uniform distribution
(Tel Aviv University)
Thursday, February 10, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 10, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 11, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The study of uniform distribution of sequences is more than a century old, with pioneering work by Hardy and Littlewood, Weyl, van der Corput and others. More recently, the focus of research has shifted to much finer quantities, such as the distribution of nearest neighbor gaps and the pair correlation function. Examples of interesting sequences for which these quantities have been studied include the zeros of the Riemann zeta function, energy levels of quantum systems, and more. In this expository talk, I will discuss what is known about these examples and discuss the many outstanding problems that this theory has to offer.
Abstract: The study of uniform distribution of sequences is more than a century old, with pioneering work by Hardy and Littlewood, Weyl, van der Corput and others. More recently, the focus of research has shifted to much finer quantities, such as the distribution of nearest neighbor gaps and the pair correlation function. Examples of interesting sequences for which these quantities have been studied include the zeros of the Riemann zeta function, energy levels of quantum systems, and more. In this expository talk, I will discuss what is known about these examples and discuss the many outstanding problems that this theory has to offer.
Peter Humphries, L^p-norm bounds for automorphic forms
(University of Virginia)
Peter Humphries, L^p-norm bounds for automorphic forms
(University of Virginia)
Thursday, February 3, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, February 3, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, February 4, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: A major area of study in analysis involves the distribution of mass of Laplacian eigenfunctions on a Riemannian manifold. A key result towards this is explicit L^p-norm bounds for Laplacian eigenfunctions in terms of their Laplacian eigenvalue, due to Sogge in 1988. Sogge's bounds are sharp on the sphere, but need not be sharp on other manifolds. I will discuss some aspects of this problem for the modular surface; in this setting, the Laplacian eigenfunctions are automorphic forms, and certain L^p-norms can be shown to be closely related to certain mixed moments of L-functions. This is joint with with Rizwanur Khan.
Abstract: A major area of study in analysis involves the distribution of mass of Laplacian eigenfunctions on a Riemannian manifold. A key result towards this is explicit L^p-norm bounds for Laplacian eigenfunctions in terms of their Laplacian eigenvalue, due to Sogge in 1988. Sogge's bounds are sharp on the sphere, but need not be sharp on other manifolds. I will discuss some aspects of this problem for the modular surface; in this setting, the Laplacian eigenfunctions are automorphic forms, and certain L^p-norms can be shown to be closely related to certain mixed moments of L-functions. This is joint with with Rizwanur Khan.
Larry Guth, Reflections on the proof(s) of the Vinogradov mean value conjecture
(MIT)
Larry Guth, Reflections on the proof(s) of the Vinogradov mean value conjecture
(MIT)
Thursday, January 27, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 28, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 27, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 28, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The Vinogradov mean value conjecture concerns the number of solutions of a system of diophantine equations. This number of solutions can also be written as a certain moment of a trigonometric polynomial. The conjecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley, and recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The details of each proof involve some intricate estimates. The goal of the talk is to try to reflect on the proof(s) in a big picture way. A key ingredient in all the proofs is to combine estimates at many different scales, usually by doing induction on scales. Why does this multi-scale induction help? What can multi-scale induction tell us and what are its limitations?
Abstract: The Vinogradov mean value conjecture concerns the number of solutions of a system of diophantine equations. This number of solutions can also be written as a certain moment of a trigonometric polynomial. The conjecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley, and recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The details of each proof involve some intricate estimates. The goal of the talk is to try to reflect on the proof(s) in a big picture way. A key ingredient in all the proofs is to combine estimates at many different scales, usually by doing induction on scales. Why does this multi-scale induction help? What can multi-scale induction tell us and what are its limitations?
Jozsef Solymosi, Rank of matrices with entries from a multiplicative group
(University of British Columbia)
Jozsef Solymosi, Rank of matrices with entries from a multiplicative group
(University of British Columbia)
Thursday, January 20, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 21, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 20, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 21, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets. (Joint work with Noga Alon)
Abstract: We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets. (Joint work with Noga Alon)
Péter Varjú, Irreducibility of random polynomials
(University of Cambridge)
Péter Varjú, Irreducibility of random polynomials
(University of Cambridge)
Thursday, January 13, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 14, 2022 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 13, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, January 14, 2022 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Consider random polynomials of degree d whose leading and constant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts that such a polynomial is irreducible in Z[x] with high probability as d grows. This conjecture is still open, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the method of proof of this result and will discuss later developments that apply this method to other models of random polynomials.
Abstract: Consider random polynomials of degree d whose leading and constant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts that such a polynomial is irreducible in Z[x] with high probability as d grows. This conjecture is still open, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the method of proof of this result and will discuss later developments that apply this method to other models of random polynomials.
Sarah Zerbes, Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces
(University College London, UK)
Sarah Zerbes, Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces
(University College London, UK)
Thursday, December 16, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 17, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 16, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 17, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Euler systems are one of the most powerful tools for proving cases of the Bloch--Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture.
Abstract: Euler systems are one of the most powerful tools for proving cases of the Bloch--Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture.
I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.
I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.
Samir Siksek, The Fermat equation and the unit equation
(University of Warwick)
Samir Siksek, The Fermat equation and the unit equation
(University of Warwick)
Thursday, December 9, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 10, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 9, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 10, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The asymptotic Fermat conjecture (AFC) states that for a number field K, and for sufficiently large primes p, the only solutions to the Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch recent work that connects the Fermat equation to the far more elementary unit equation, and explain how this surprising connection can be exploited to prove AFC for several infinite families of number fields. This talk is based on joint work with Nuno Freitas, Alain Kraus and Haluk Sengun.
Abstract: The asymptotic Fermat conjecture (AFC) states that for a number field K, and for sufficiently large primes p, the only solutions to the Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch recent work that connects the Fermat equation to the far more elementary unit equation, and explain how this surprising connection can be exploited to prove AFC for several infinite families of number fields. This talk is based on joint work with Nuno Freitas, Alain Kraus and Haluk Sengun.
Kiran Kedlaya, Orders of abelian varieties over $\mathbb{F}_2$
(University of California San Diego)
Kiran Kedlaya, Orders of abelian varieties over $\mathbb{F}_2$
(University of California San Diego)
Thursday, December 2, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 3, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 2, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, December 3, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.
Abstract: We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.
Alexei Skorobogatov, On uniformity conjectures for abelian varieties and K3 surfaces
(Imperial College London)
Alexei Skorobogatov, On uniformity conjectures for abelian varieties and K3 surfaces
(Imperial College London)
Thursday, November 25, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 26, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 25, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 26, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Néron–Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group. The talk is based on a joint work with Martin Orr and Yuri Zarhin.
Abstract: I will discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Néron–Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group. The talk is based on a joint work with Martin Orr and Yuri Zarhin.
Myrto Mavraki, Towards uniformity in the dynamical Bogomolov conjecture
(Harvard University)
Myrto Mavraki, Towards uniformity in the dynamical Bogomolov conjecture
(Harvard University)
Thursday, November 18, 2021 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 19, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 18, 2021 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 19, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.
Abstract: Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.
In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end, we establish an instance of a 'relative dynamical Bogomolov'. This is work in progress joint with Harry Schmidt (University of Basel).
In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end, we establish an instance of a 'relative dynamical Bogomolov'. This is work in progress joint with Harry Schmidt (University of Basel).
Avi Wigderson, Randomness
(Institute for Advanced Study)
Avi Wigderson, Randomness
(Institute for Advanced Study)
Thursday, November 11, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 12, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 11, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 12, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference between the two?
Abstract: Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference between the two?
Humanity has pondered the meaning and utility of randomness for millennia.
Humanity has pondered the meaning and utility of randomness for millennia.
There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable! Which of these applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from somewhat "unpredictable" phenomena like the weather or the stock market?
There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable! Which of these applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from somewhat "unpredictable" phenomena like the weather or the stock market?
A computational theory of randomness, developed in the past several decades, reveals (perhaps counter-intuitively) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory, notions of pseudo-randomness, and connections to computational intractability.
A computational theory of randomness, developed in the past several decades, reveals (perhaps counter-intuitively) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory, notions of pseudo-randomness, and connections to computational intractability.
It is interesting that Number Theory played an important role throughout this development. It supplied problems whose algorithmic solution make randomness seem powerful, problems for which randomness can be eliminated from such solutions, and problems where the power of randomness remains a major challenge for computational complexity theorists and mathematicians. I will use these problems (and others) to demonstrate aspects of this theory.
It is interesting that Number Theory played an important role throughout this development. It supplied problems whose algorithmic solution make randomness seem powerful, problems for which randomness can be eliminated from such solutions, and problems where the power of randomness remains a major challenge for computational complexity theorists and mathematicians. I will use these problems (and others) to demonstrate aspects of this theory.
Katherine Stange, Algebraic Number Starscapes
(University of Colorado, Boulder)
Katherine Stange, Algebraic Number Starscapes
(University of Colorado, Boulder)
Thursday, November 4, 2021 (9am PDT, 12pm EDT, 4pm BST, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 5, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, November 4, 2021 (9am PDT, 12pm EDT, 4pm BST, 5pm CET, 6pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, November 5, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In the spirit of experimentation, at the Fall 2019 ICERM special semester on “Illustrating Mathematics,” I began drawing algebraic numbers in the complex plane. Edmund Harriss, Steve Trettel and I sized the numbers by arithmetic complexity and found a wealth of pattern and structure. In this talk, I’ll take you on a visual tour and share some of the mathematical explanations we found for what can be quite stunning pictures (in the hands of a mathematician and artist like Edmund). This experience gave me a new perspective on complex Diophantine approximation: one can view approximation properties as being dictated by the geometry of the map from coefficient space to root space in different polynomial degrees. I’ll explain this geometry, and discuss a few Diophantine results, known and new, in this context.
Abstract: In the spirit of experimentation, at the Fall 2019 ICERM special semester on “Illustrating Mathematics,” I began drawing algebraic numbers in the complex plane. Edmund Harriss, Steve Trettel and I sized the numbers by arithmetic complexity and found a wealth of pattern and structure. In this talk, I’ll take you on a visual tour and share some of the mathematical explanations we found for what can be quite stunning pictures (in the hands of a mathematician and artist like Edmund). This experience gave me a new perspective on complex Diophantine approximation: one can view approximation properties as being dictated by the geometry of the map from coefficient space to root space in different polynomial degrees. I’ll explain this geometry, and discuss a few Diophantine results, known and new, in this context.
Dimitris Koukoulopoulos, Towards a high-dimensional theory of divisors of integers
(University of Montreal)
Dimitris Koukoulopoulos, Towards a high-dimensional theory of divisors of integers
(University of Montreal)
Thursday, October 28, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 29, 2021 (2am AEDT, 4am NZDT)
Thursday, October 28, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 29, 2021 (2am AEDT, 4am NZDT)
Abstract: In this talk, I will survey some results about high-dimensional phenomena in the theory of divisors of integers.
Abstract: In this talk, I will survey some results about high-dimensional phenomena in the theory of divisors of integers.
Fix an integer $k\ge2$ and pick an integer $n\le x$ uniformly at random. We then consider the following two basic problems:
Fix an integer $k\ge2$ and pick an integer $n\le x$ uniformly at random. We then consider the following two basic problems:
What are the chances that $n$ can be factored as $n=d_1\cdots d_k$ with each factor $d_i$ lying in some prescribed dyadic interval $[y_i,2y_i]$?
What are the chances that $n$ can be factored as $n=d_1\cdots d_k$ with each factor $d_i$ lying in some prescribed dyadic interval $[y_i,2y_i]$?
What are the chances that we can find $k$ divisors of $n$, say $d_1,\dots,d_k$, such that $|\log(d_j/d_i)|<1$ for all $i,j$, and which are all composed from a prescribed set of prime factors of $n$?
What are the chances that we can find $k$ divisors of $n$, say $d_1,\dots,d_k$, such that $|\log(d_j/d_i)|<1$ for all $i,j$, and which are all composed from a prescribed set of prime factors of $n$?
The first problem is a high-dimensional generalization of the Erd\H os multiplication table problem; it is well-understood when $k\le 6$, but less so when $k\ge7$. The second problem is related to Hooley’s function $\Delta(n):=\max_u \#\{d|n:u<\log d\le u+1\}$ that measures the concentration of the sequence of divisors of $n$, and that has surprising applications to Diophantine number theory.
The first problem is a high-dimensional generalization of the Erd\H os multiplication table problem; it is well-understood when $k\le 6$, but less so when $k\ge7$. The second problem is related to Hooley’s function $\Delta(n):=\max_u \#\{d|n:u<\log d\le u+1\}$ that measures the concentration of the sequence of divisors of $n$, and that has surprising applications to Diophantine number theory.
In recent work with Kevin Ford and Ben Green, we built on the earlier work on Problem 1 to develop a new approach to Problem 2. This led to an improved lower bound on the almost-sure behaviour of Hooley’s $\Delta$-function, that we conjecture to be optimal. The new ideas might in turn shed light to Problem 1 and other high-dimensional phenomena about divisors of integers.
In recent work with Kevin Ford and Ben Green, we built on the earlier work on Problem 1 to develop a new approach to Problem 2. This led to an improved lower bound on the almost-sure behaviour of Hooley’s $\Delta$-function, that we conjecture to be optimal. The new ideas might in turn shed light to Problem 1 and other high-dimensional phenomena about divisors of integers.
Johan Commelin, Liquid Tensor Experiment
(Albert–Ludwigs-Universität Freiburg)
Johan Commelin, Liquid Tensor Experiment
(Albert–Ludwigs-Universität Freiburg)
Thursday, October 21, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 22, 2021 (2am AEST, 4am NZST)
Thursday, October 21, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 22, 2021 (2am AEST, 4am NZST)
Abstract: In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid $\mathbb{R}$-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that in a couple of months we will have completed the full challenge.
Abstract: In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid $\mathbb{R}$-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that in a couple of months we will have completed the full challenge.
In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.
In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.
Jean-Marc Deshouillers, Are factorials sums of three cubes?
(Institut de Mathématiques de Bordeaux)
Jean-Marc Deshouillers, Are factorials sums of three cubes?
(Institut de Mathématiques de Bordeaux)
Thursday, October 14, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 15, 2021 (2am AEST, 4am NZST)
Thursday, October 14, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 15, 2021 (2am AEST, 4am NZST)
Henryk Iwaniec, Remarks on the large sieve
(Rutgers University)
Henryk Iwaniec, Remarks on the large sieve
(Rutgers University)
A talk in honor of John Friedlander's 80th birthday
Special Chairs: Leo Goldmakher (Williams College) and Andrew Granville (University of Montreal)
Special Chairs: Leo Goldmakher (Williams College) and Andrew Granville (University of Montreal)
Thursday, October 7, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 8, 2021 (2am AEST, 4am NZST)
Thursday, October 7, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 8, 2021 (2am AEST, 4am NZST)
Abstract: The concept of the large sieve will be discussed in various contexts. The power and limitation of basic estimates will be illustrated with some examples. Recent work on the large sieve for characters to prime moduli will be explained.
Abstract: The concept of the large sieve will be discussed in various contexts. The power and limitation of basic estimates will be illustrated with some examples. Recent work on the large sieve for characters to prime moduli will be explained.
Anish Ghosh, Values of quadratic forms at integer points
(Tata Institute of Fundamental Research)
Anish Ghosh, Values of quadratic forms at integer points
(Tata Institute of Fundamental Research)
Thursday, September 30, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 1, 2021 (1am AEST, 4am NZDT)
Thursday, September 30, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 1, 2021 (1am AEST, 4am NZDT)
Abstract: A famous theorem of Margulis, resolving a conjecture of Oppenheim, states that an indefinite, irrational quadratic form in at least three variables takes a dense set of values at integer points. Recently there has been a push towards establishing effective versions of Margulis's theorem. I will explain Margulis's approach to this problem which involves the ergodic theory of group actions on homogeneous spaces. I will then discuss some new effective results in this direction. These results use a variety of techniques including tools from ergodic theory, analytic number theory as well as the geometry of numbers.
Abstract: A famous theorem of Margulis, resolving a conjecture of Oppenheim, states that an indefinite, irrational quadratic form in at least three variables takes a dense set of values at integer points. Recently there has been a push towards establishing effective versions of Margulis's theorem. I will explain Margulis's approach to this problem which involves the ergodic theory of group actions on homogeneous spaces. I will then discuss some new effective results in this direction. These results use a variety of techniques including tools from ergodic theory, analytic number theory as well as the geometry of numbers.
Alina Carmen Cojocaru, Bounds for the distribution of the Frobenius traces associated to abelian varieties
(University of Illinois at Chicago and Institute of Mathematics of the Romanian Academy)
Alina Carmen Cojocaru, Bounds for the distribution of the Frobenius traces associated to abelian varieties
(University of Illinois at Chicago and Institute of Mathematics of the Romanian Academy)
Thursday, September 23, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 24, 2021 (1am AEST, 3am NZST)
Thursday, September 23, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 24, 2021 (1am AEST, 3am NZST)
Abstract: In 1976, Serge Lang and Hale Trotter conjectured the asymptotic growth of the number $\pi_A(x, t)$ of primes $p < x$ for which the Frobenius trace $a_p$ of a non-CM elliptic curve $A/\mathbb{Q}$ equals an integer $t$. Even though their conjecture remains open, over the past decades the study of the counting function $\pi_A(x, t)$ has witnessed remarkable advances. We will discuss generalizations of such studies in the setting of an abelian variety $A/\mathbb{Q}$ of arbitrary dimension and we will present non-trivial upper bounds for the corresponding counting function $\pi_A(x, t)$. This is joint work with Tian Wang (University of Illinois at Chicago).
Abstract: In 1976, Serge Lang and Hale Trotter conjectured the asymptotic growth of the number $\pi_A(x, t)$ of primes $p < x$ for which the Frobenius trace $a_p$ of a non-CM elliptic curve $A/\mathbb{Q}$ equals an integer $t$. Even though their conjecture remains open, over the past decades the study of the counting function $\pi_A(x, t)$ has witnessed remarkable advances. We will discuss generalizations of such studies in the setting of an abelian variety $A/\mathbb{Q}$ of arbitrary dimension and we will present non-trivial upper bounds for the corresponding counting function $\pi_A(x, t)$. This is joint work with Tian Wang (University of Illinois at Chicago).
Martín Sombra, The mean height of the solution set of a system of polynomial equations
(ICREA and University of Barcelona)
Martín Sombra, The mean height of the solution set of a system of polynomial equations
(ICREA and University of Barcelona)
Thursday, September 16, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 17, 2021 (1am AEST, 3am NZST)
Thursday, September 16, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 17, 2021 (1am AEST, 3am NZST)
Abstract: Bernstein’s theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, we can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.
Abstract: Bernstein’s theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, we can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.
Emmanuel Kowalski, Harmonic analysis over finite fields and equidistribution
(ETH Zürich)
Emmanuel Kowalski, Harmonic analysis over finite fields and equidistribution
(ETH Zürich)
Thursday, September 9, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 10, 2021 (1am AEST, 3am NZST)
Thursday, September 9, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 10, 2021 (1am AEST, 3am NZST)
Abstract: In 1976, Deligne defined a geometric version of the Fourier transform over finite fields, leading to significant applications in number theory.
Abstract: In 1976, Deligne defined a geometric version of the Fourier transform over finite fields, leading to significant applications in number theory.
For a number of applications, including equidistribution of exponential sums parameterized by multiplicative characters, it would be very helpful to have a similar geometric harmonic analysis for other groups. I will discuss ongoing joint work with A. Forey and J. Fresán in which we establish some results in this direction by generalizing ideas of Katz. I will present the general equidistribution theorem for exponential sums parameterized by characters that we obtain, and discuss applications, as well as challenges, open questions and mysteries.
For a number of applications, including equidistribution of exponential sums parameterized by multiplicative characters, it would be very helpful to have a similar geometric harmonic analysis for other groups. I will discuss ongoing joint work with A. Forey and J. Fresán in which we establish some results in this direction by generalizing ideas of Katz. I will present the general equidistribution theorem for exponential sums parameterized by characters that we obtain, and discuss applications, as well as challenges, open questions and mysteries.
Lars Kühne, The uniform Bogomolov conjecture for algebraic curves
(University of Copenhagen)
Lars Kühne, The uniform Bogomolov conjecture for algebraic curves
(University of Copenhagen)
Thursday, September 2, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 3, 2021 (1am AEST, 3am NZST)
Thursday, September 2, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 3, 2021 (1am AEST, 3am NZST)
Abstract: I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces, but it also follows from independent work by Yuan and Zhang, which has been recently reported in this seminar. I will therefore focus on the application that motivated my work, namely a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell-Lang conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will restrict to the case of curves for simplicity.
Abstract: I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces, but it also follows from independent work by Yuan and Zhang, which has been recently reported in this seminar. I will therefore focus on the application that motivated my work, namely a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell-Lang conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will restrict to the case of curves for simplicity.
Alexandru Zaharescu, Some remarks on Landau - Siegel zeros
(University of Illinois at Urbana-Champaign)
Alexandru Zaharescu, Some remarks on Landau - Siegel zeros
(University of Illinois at Urbana-Champaign)
Thursday, August 26, 2021 (2pm PDT, 5pm EDT, 10pm BST, 11pm CEST)
Friday, August 27, 2021 (12am Israel Daylight Time, 2:30am Indian Standard Time, 5am CST, 7am AEST, 9am NZST)
Thursday, August 26, 2021 (2pm PDT, 5pm EDT, 10pm BST, 11pm CEST)
Friday, August 27, 2021 (12am Israel Daylight Time, 2:30am Indian Standard Time, 5am CST, 7am AEST, 9am NZST)
Abstract: In the first part of the talk I will survey some known results related to the hypothetical existence of Landau - Siegel zeros. In the second part of the talk I will discuss some recent joint work with Hung Bui and Kyle Pratt in which we show that the existence of Landau - Siegel zeros has implications for the behavior of L - functions at the central point.
Abstract: In the first part of the talk I will survey some known results related to the hypothetical existence of Landau - Siegel zeros. In the second part of the talk I will discuss some recent joint work with Hung Bui and Kyle Pratt in which we show that the existence of Landau - Siegel zeros has implications for the behavior of L - functions at the central point.
Zeev Dvir, The Kakeya set conjecture over rings of integers modulo square free m
(Princeton University)
Zeev Dvir, The Kakeya set conjecture over rings of integers modulo square free m
(Princeton University)
Thursday, August 19, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 20, 2021 (1am AEST, 3am NZST)
Thursday, August 19, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 20, 2021 (1am AEST, 3am NZST)
Abstract: We show that, when N is any square-free integer, the size of the smallest Kakeya set in (ℤ/Nℤ)^n is at least C_{eps,n}*N^{n-eps} for any eps>0 -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the p-rank of the incidence matrix of points and hyperplanes over (ℤ/p^kℤ)^n. Joint work with Manik Dhar.
Abstract: We show that, when N is any square-free integer, the size of the smallest Kakeya set in (ℤ/Nℤ)^n is at least C_{eps,n}*N^{n-eps} for any eps>0 -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the p-rank of the incidence matrix of points and hyperplanes over (ℤ/p^kℤ)^n. Joint work with Manik Dhar.
Francesco Amoroso, Bounded Height in Pencils of Subgroups of finite rank
(University of Caen)
Francesco Amoroso, Bounded Height in Pencils of Subgroups of finite rank
(University of Caen)
Thursday, August 12, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 13, 2021 (1am AEST, 3am NZST)
Thursday, August 12, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 13, 2021 (1am AEST, 3am NZST)
Abstract: [Joint work with D.Masser and U.Zannier]
Abstract: [Joint work with D.Masser and U.Zannier]
Let n>1 be a varying natural number. By a result of Beukers, the solutions of t^n+(1-t)^n=1 have uniformly bounded height. What happens if we allow rational exponents?
Let n>1 be a varying natural number. By a result of Beukers, the solutions of t^n+(1-t)^n=1 have uniformly bounded height. What happens if we allow rational exponents?
We consider the analogous question replacing the affine curve x+y=1 with an arbitrary irreducible curve and {t^n | n rational} with the division group of a finitely generated subgroup.
We consider the analogous question replacing the affine curve x+y=1 with an arbitrary irreducible curve and {t^n | n rational} with the division group of a finitely generated subgroup.
Frank Calegari, Digits
(University of Chicago)
Frank Calegari, Digits
(University of Chicago)
Thursday, August 5, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 6, 2021 (1am AEST, 3am NZST)
Thursday, August 5, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, August 6, 2021 (1am AEST, 3am NZST)
Abstract: We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This is work in progress with Soundararajan which we may well write up one day.
Abstract: We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This is work in progress with Soundararajan which we may well write up one day.
Arno Fehm, Is Z diophantine in Q?
(Technische Universität Dresden)
Arno Fehm, Is Z diophantine in Q?
(Technische Universität Dresden)
Thursday, July 29, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 30, 2021 (1am AEST, 3am NZST)
Thursday, July 29, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 30, 2021 (1am AEST, 3am NZST)
Abstract: Are the integers the projection of the rational zeros of a polynomial in several variables onto the first coordinate? The aim of this talk is to motivate and discuss this longstanding question. I will survey some results regarding diophantine sets and Hilbert's tenth problem (the existence of an algorithm that decides whether a polynomial has a zero) in fields and will discuss a few conjectures, some classical and some more recent, that suggest that the answer to the question should be negative.
Abstract: Are the integers the projection of the rational zeros of a polynomial in several variables onto the first coordinate? The aim of this talk is to motivate and discuss this longstanding question. I will survey some results regarding diophantine sets and Hilbert's tenth problem (the existence of an algorithm that decides whether a polynomial has a zero) in fields and will discuss a few conjectures, some classical and some more recent, that suggest that the answer to the question should be negative.
Kumar Murty, Periods and Mixed Motives
(University of Toronto)
Kumar Murty, Periods and Mixed Motives
(University of Toronto)
Thursday, July 22, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 23, 2021 (1am AEST, 3am NZST)
Thursday, July 22, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 23, 2021 (1am AEST, 3am NZST)
Abstract: We discuss some consequences of Grothendieck's Period Conjecture in the context of mixed motives. In particular, this conjecture implies that zeta(3), log 2 and pi are algebraically independent (contrary to an expectation of Euler). After some 'motivation' and introductory remarks on periods, we derive our consequences as a result of studying mixed motives whose Galois group has a large unipotent radical. This is joint work with Payman Eskandari.
Abstract: We discuss some consequences of Grothendieck's Period Conjecture in the context of mixed motives. In particular, this conjecture implies that zeta(3), log 2 and pi are algebraically independent (contrary to an expectation of Euler). After some 'motivation' and introductory remarks on periods, we derive our consequences as a result of studying mixed motives whose Galois group has a large unipotent radical. This is joint work with Payman Eskandari.
Ricardo Menares, p-adic distribution of CM points
(Pontificia Universidad Católica de Chile)
Ricardo Menares, p-adic distribution of CM points
(Pontificia Universidad Católica de Chile)
Thursday, July 15, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 16, 2021 (1am AEST, 3am NZST)
Thursday, July 15, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 16, 2021 (1am AEST, 3am NZST)
Abstract: CM points are the isomorphism classes of CM elliptic curves. When ordered by the absolute value of the discriminant of the endomorphism ring, CM points are distributed along the complex (level one) modular curve according to the hyperbolic measure. This statement was proved by Duke for fundamental discriminants and later, building on this work, Clozel and Ullmo proved it in full generality.
Abstract: CM points are the isomorphism classes of CM elliptic curves. When ordered by the absolute value of the discriminant of the endomorphism ring, CM points are distributed along the complex (level one) modular curve according to the hyperbolic measure. This statement was proved by Duke for fundamental discriminants and later, building on this work, Clozel and Ullmo proved it in full generality.
In this talk, we establish the p-adic analogue of this result. Namely, for a fixed prime p we regard the CM points as a subset of the p-adic space attached to the modular curve and we classify the possible accumulation measures of CM points as the discriminant varies. In particular, we find that there are infinitely many such measures. This is in stark contrast to the complex case, where the hyperbolic measure is the unique accumulation measure.
In this talk, we establish the p-adic analogue of this result. Namely, for a fixed prime p we regard the CM points as a subset of the p-adic space attached to the modular curve and we classify the possible accumulation measures of CM points as the discriminant varies. In particular, we find that there are infinitely many such measures. This is in stark contrast to the complex case, where the hyperbolic measure is the unique accumulation measure.
As an application, we show that for any finite set S of prime numbers, the set of singular moduli which are S-units is finite.
As an application, we show that for any finite set S of prime numbers, the set of singular moduli which are S-units is finite.
This is joint work with Sebastián Herrero (PUC Valparaíso) and Juan Rivera-Letelier (Rochester).
This is joint work with Sebastián Herrero (PUC Valparaíso) and Juan Rivera-Letelier (Rochester).
Brian Conrey, Moments, ratios, arithmetic functions in short intervals and random matrix averages
(American Institute of Mathematics)
Brian Conrey, Moments, ratios, arithmetic functions in short intervals and random matrix averages
(American Institute of Mathematics)
Thursday, July 8, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 9, 2021 (1am AEST, 3am NZST)
Thursday, July 8, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 9, 2021 (1am AEST, 3am NZST)
Abstract: We discuss how the conjectures for moments of L-functions imply short interval averages of the L-coefficient convolutions. Similarly the ratios conjectures lead to short interval averages of the convolutions of coefficients at almost primes. These in turn are related to random matrix averages considered by Diaconis - Gamburd and by Diaconis - Shahshahani.
Abstract: We discuss how the conjectures for moments of L-functions imply short interval averages of the L-coefficient convolutions. Similarly the ratios conjectures lead to short interval averages of the convolutions of coefficients at almost primes. These in turn are related to random matrix averages considered by Diaconis - Gamburd and by Diaconis - Shahshahani.
Manjul Bhargava, Galois groups of random integer polynomials
(Princeton University)
Manjul Bhargava, Galois groups of random integer polynomials
(Princeton University)
A talk in honor of Don Zagier's 70th birthday
Special Chair: Pieter Moree (Max Planck Institute for Mathematics)
Special Chair: Pieter Moree (Max Planck Institute for Mathematics)
Thursday, July 1, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 2, 2021 (1am AEST, 3am NZST)
Thursday, July 1, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 2, 2021 (1am AEST, 3am NZST)
Abstract: Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we prove the "Weak van der Waerden Conjecture", which states that the number of such polynomials is $O_\epsilon(H^{n-1+\epsilon})$, for all degrees $n$.
Abstract: Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we prove the "Weak van der Waerden Conjecture", which states that the number of such polynomials is $O_\epsilon(H^{n-1+\epsilon})$, for all degrees $n$.
Annette Huber-Klawitter, Periods and O-minimality
(University of Freiburg)
Annette Huber-Klawitter, Periods and O-minimality
(University of Freiburg)
Thursday, June 24, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 25, 2021 (1am AEST, 3am NZST)
Thursday, June 24, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 25, 2021 (1am AEST, 3am NZST)
Abstract: Roughly, periods are numbers obtained by integrating algebraic differential forms over domains of integration also of arithmetic nature. I am going to give a survey on the state of the period conjecture and different points of view. I also want to present a relation to o-minimal geometry.
Abstract: Roughly, periods are numbers obtained by integrating algebraic differential forms over domains of integration also of arithmetic nature. I am going to give a survey on the state of the period conjecture and different points of view. I also want to present a relation to o-minimal geometry.
Shou-Wu Zhang, Adelic line bundles over quasi-projective varieties
(Princeton University)
Shou-Wu Zhang, Adelic line bundles over quasi-projective varieties
(Princeton University)
Thursday, June 17, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 18, 2021 (1am AEST, 3am NZST)
Thursday, June 17, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 18, 2021 (1am AEST, 3am NZST)
Abstract: For quasi-projective varieties over finitely generated fields, we develop a theory of adelic line bundles including an equidistribution theorem for Galois orbits of small points. In this lecture, we will explain this theory and its application to arithmetic of abelian varieties, dynamical systems, and their moduli. This is a joint work with Xinyi Yuan.
Abstract: For quasi-projective varieties over finitely generated fields, we develop a theory of adelic line bundles including an equidistribution theorem for Galois orbits of small points. In this lecture, we will explain this theory and its application to arithmetic of abelian varieties, dynamical systems, and their moduli. This is a joint work with Xinyi Yuan.
Matthew Young, The Weyl bound for Dirichlet L-functions
(Texas A&M University)
Matthew Young, The Weyl bound for Dirichlet L-functions
(Texas A&M University)
Thursday, June 10, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 11, 2021 (1am AEST, 3am NZST)
Thursday, June 10, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 11, 2021 (1am AEST, 3am NZST)
Abstract: There is an analogy between the behavior of the Riemann zeta function high in the critical strip, and the behavior of Dirichlet L-functions of large conductors. In many important problems, our understanding of Dirichlet L-functions is weaker than for zeta; for example, the zero-free regions are not of the same quality due to the possible Landau-Siegel zero. This talk will discuss recent progress (joint with Ian Petrow) on subconvexity bounds for Dirichlet L-functions. These new bounds now match the original subconvexity bound for the zeta function derived by Hardy and Littlewood using Weyl's differencing method.
Abstract: There is an analogy between the behavior of the Riemann zeta function high in the critical strip, and the behavior of Dirichlet L-functions of large conductors. In many important problems, our understanding of Dirichlet L-functions is weaker than for zeta; for example, the zero-free regions are not of the same quality due to the possible Landau-Siegel zero. This talk will discuss recent progress (joint with Ian Petrow) on subconvexity bounds for Dirichlet L-functions. These new bounds now match the original subconvexity bound for the zeta function derived by Hardy and Littlewood using Weyl's differencing method.
Antoine Chambert-Loir, From complex function theory to non-archimedean spaces - a number theoretical thread
(Université Paris-Diderot)
Antoine Chambert-Loir, From complex function theory to non-archimedean spaces - a number theoretical thread
(Université Paris-Diderot)
Thursday, June 3, 2021 (11am PDT, 2pm EDT, 7pm BST, 8pm CEST, 9pm Israel Daylight Time, 11:30pm Indian Standard Time)
Friday, June 4, 2021 (2am CST, 4am AEST, 6am NZST)
Thursday, June 3, 2021 (11am PDT, 2pm EDT, 7pm BST, 8pm CEST, 9pm Israel Daylight Time, 11:30pm Indian Standard Time)
Friday, June 4, 2021 (2am CST, 4am AEST, 6am NZST)
Abstract: Diophantine geometry and complex function theory have a long and well known history of mutual friendship, attested, for example, by the fruitful interactions between height functions and potential theory. In the last 50 years, interactions even deepened with the invention of Arakelov geometry (Arakelov, Gillet/Soulé, Faltings) and its application by Szpiro/Ullmo/Zhang to equidistribution theorems and the Bogomolov conjecture. Roughly at the same time, Berkovich invented a new kind of non-archimedean analytic spaces which possess a rich and well behaved geometric structure. This opened the way to non-archimedean potential theory (Baker/Rumely, Favre/Rivera-Letelier), or to arithmetic/geometric equidistribution theorems in this case. More recently, Ducros and myself introduced basic ideas from tropical geometry and a construction of Lagerberg to construct a calculus of (p,q)-forms on Berkovich spaces, which is an analogue of the corresponding calculus on complex manifolds, and seems to be an attractive candidate for being the p-adic side of height function theory.
Abstract: Diophantine geometry and complex function theory have a long and well known history of mutual friendship, attested, for example, by the fruitful interactions between height functions and potential theory. In the last 50 years, interactions even deepened with the invention of Arakelov geometry (Arakelov, Gillet/Soulé, Faltings) and its application by Szpiro/Ullmo/Zhang to equidistribution theorems and the Bogomolov conjecture. Roughly at the same time, Berkovich invented a new kind of non-archimedean analytic spaces which possess a rich and well behaved geometric structure. This opened the way to non-archimedean potential theory (Baker/Rumely, Favre/Rivera-Letelier), or to arithmetic/geometric equidistribution theorems in this case. More recently, Ducros and myself introduced basic ideas from tropical geometry and a construction of Lagerberg to construct a calculus of (p,q)-forms on Berkovich spaces, which is an analogue of the corresponding calculus on complex manifolds, and seems to be an attractive candidate for being the p-adic side of height function theory.
Robert Tichy, Equidistribution, exponential sums and van der Corput sets
(TU Graz)
Robert Tichy, Equidistribution, exponential sums and van der Corput sets
(TU Graz)
Thursday, May 27, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 28, 2021 (1am AEST, 3am NZST)
Thursday, May 27, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 28, 2021 (1am AEST, 3am NZST)
Abstract: The talk starts with a survey on Sarkoezy`s results on difference sets and with Furstenberg`s dynamic approach to additive problems. We present some results of a joint work with Bergelson, Kolesnik, Son and Madritsch concerning multidimensional van der Corput sets based on new bounds for exponential sums. In a second part we give a brief introduction on equidistribution theory focusing on the interplay of exponential sums with difference theorems. In a third part Hardy fields are discussed in some detail. This concept was introduced to equidistribution theory by Boshernitzan and it tuned out to be very fruitful. We will report on recent results of Bergelson et al. and at the very end on applications to diophantine approximation. This includes results concerning the approximation of polynomial-like functions along primes which were established in a joint work with Madritsch and sharpened very recently by my PhD student Minelli.
Abstract: The talk starts with a survey on Sarkoezy`s results on difference sets and with Furstenberg`s dynamic approach to additive problems. We present some results of a joint work with Bergelson, Kolesnik, Son and Madritsch concerning multidimensional van der Corput sets based on new bounds for exponential sums. In a second part we give a brief introduction on equidistribution theory focusing on the interplay of exponential sums with difference theorems. In a third part Hardy fields are discussed in some detail. This concept was introduced to equidistribution theory by Boshernitzan and it tuned out to be very fruitful. We will report on recent results of Bergelson et al. and at the very end on applications to diophantine approximation. This includes results concerning the approximation of polynomial-like functions along primes which were established in a joint work with Madritsch and sharpened very recently by my PhD student Minelli.
Alice Silverberg, Cryptographic Multilinear Maps and Miscellaneous Musings
(University of California, Irvine)
Alice Silverberg, Cryptographic Multilinear Maps and Miscellaneous Musings
(University of California, Irvine)
Thursday, May 20, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 21, 2021 (1am AEST, 3am NZST)
Thursday, May 20, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 21, 2021 (1am AEST, 3am NZST)
Abstract: Recognizing that many of us have Zoom fatigue, I will keep this talk light, without too many technical details. In addition to discussing an open problem on multilinear maps that has applications to cryptography, I'll give miscellaneous musings about things I've learned over the years that I wish I'd learned sooner.
Abstract: Recognizing that many of us have Zoom fatigue, I will keep this talk light, without too many technical details. In addition to discussing an open problem on multilinear maps that has applications to cryptography, I'll give miscellaneous musings about things I've learned over the years that I wish I'd learned sooner.
Alex Kontorovich, Arithmetic Groups and Sphere Packings
(Rutgers University)
Alex Kontorovich, Arithmetic Groups and Sphere Packings
(Rutgers University)
Thursday, May 13, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 14, 2021 (1am AEST, 3am NZST)
Thursday, May 13, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 14, 2021 (1am AEST, 3am NZST)
Abstract: We discuss recent progress on understanding connections between the objects in the title.
Abstract: We discuss recent progress on understanding connections between the objects in the title.
Akshay Venkatesh, A brief history of Hecke operators
(Institute for Advanced Study)
Akshay Venkatesh, A brief history of Hecke operators
(Institute for Advanced Study)
Thursday, May 6, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 7, 2021 (1am AEST, 3am NZST)
Thursday, May 6, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 7, 2021 (1am AEST, 3am NZST)
Abstract: This is an expository lecture about Hecke operators, in the context of number theory. We will trace some of the history of the ideas, starting before Hecke's birth and proceeding through the subsequent century. In particular we will discuss some of the original motivations and then the impact of ideas from representation theory and algebraic geometry. This lecture is aimed at non-experts.
Abstract: This is an expository lecture about Hecke operators, in the context of number theory. We will trace some of the history of the ideas, starting before Hecke's birth and proceeding through the subsequent century. In particular we will discuss some of the original motivations and then the impact of ideas from representation theory and algebraic geometry. This lecture is aimed at non-experts.
Pietro Corvaja, On the local-to-global principle for value sets
(University of Udine)
Pietro Corvaja, On the local-to-global principle for value sets
(University of Udine)
Special Chair: Andrew Granville (University of Montreal)
Special Chair: Andrew Granville (University of Montreal)
Thursday, April 29, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 30, 2021 (1am AEST, 3am NZST)
Thursday, April 29, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 30, 2021 (1am AEST, 3am NZST)
Abstract: Given a finite morphism f: X -> Y between algebraic curves over number fields, we study the set of rational (or integral) points in Y having a pre-image in every p-adic completion of the number field, but no rational pre-images. In particular, we investigate whether this set can be infinite.
Abstract: Given a finite morphism f: X -> Y between algebraic curves over number fields, we study the set of rational (or integral) points in Y having a pre-image in every p-adic completion of the number field, but no rational pre-images. In particular, we investigate whether this set can be infinite.
We will mark the 1 year anniversary of the Number Theory Web Seminar.
Renate Scheidler, Computing modular polynomials and isogeny graphs of rank 2 Drinfeld modules
(University of Calgary)
Renate Scheidler, Computing modular polynomials and isogeny graphs of rank 2 Drinfeld modules
(University of Calgary)
Thursday, April 22, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 23, 2021 (1am AEST, 3am NZST)
Thursday, April 22, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 23, 2021 (1am AEST, 3am NZST)
Abstract: Drinfeld modules represent the function field analogue of the theory of complex multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld in the 1970s in the course of proving the Langlands conjectures for GL(2) over global function fields. Drinfeld modules of rank 2 exhibit very similar behaviour to elliptic curves: they are classified as ordinary or supersingular, support isogenies and their duals, and their endomorphism rings have an analogous structure. Their isomorphism classes are parameterized by j-invariants, and Drinfeld modular polynomials can be used to compute their isogeny graphs whose ordinary connected components take the shape of volcanos. While the rich analytic and algebraic theory of Drinfeld modules has undergone extensive investigation, very little has been explored from a computational perspective. This research represents the first foray in this direction, introducing an algorithm for computing Drinfeld modular polynomials and isogeny graphs.
Abstract: Drinfeld modules represent the function field analogue of the theory of complex multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld in the 1970s in the course of proving the Langlands conjectures for GL(2) over global function fields. Drinfeld modules of rank 2 exhibit very similar behaviour to elliptic curves: they are classified as ordinary or supersingular, support isogenies and their duals, and their endomorphism rings have an analogous structure. Their isomorphism classes are parameterized by j-invariants, and Drinfeld modular polynomials can be used to compute their isogeny graphs whose ordinary connected components take the shape of volcanos. While the rich analytic and algebraic theory of Drinfeld modules has undergone extensive investigation, very little has been explored from a computational perspective. This research represents the first foray in this direction, introducing an algorithm for computing Drinfeld modular polynomials and isogeny graphs.
This is joint work with Perlas Caranay and Matt Greenberg, as well as ongoing research with Edgar Pacheco Castan. Some familiarity with elliptic curves is expected for this talk, but no prior knowledge of Drinfeld modules is assumed.
This is joint work with Perlas Caranay and Matt Greenberg, as well as ongoing research with Edgar Pacheco Castan. Some familiarity with elliptic curves is expected for this talk, but no prior knowledge of Drinfeld modules is assumed.
Jonathan Keating, Joint Moments
(University of Oxford)
Jonathan Keating, Joint Moments
(University of Oxford)
Thursday, April 15, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 16, 2021 (1am AEST, 3am NZST)
Thursday, April 15, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 16, 2021 (1am AEST, 3am NZST)
Abstract: I will discuss the joint moments of the Riemann zeta-function and its derivative, and the corresponding joint moments of the characteristic polynomials of random unitary matrices and their derivatives.
Abstract: I will discuss the joint moments of the Riemann zeta-function and its derivative, and the corresponding joint moments of the characteristic polynomials of random unitary matrices and their derivatives.
János Pintz, On the mean value of the remainder term of the prime number formula
(Alfréd Rényi Institute of Mathematics)
János Pintz, On the mean value of the remainder term of the prime number formula
(Alfréd Rényi Institute of Mathematics)
Thursday, April 8, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 9, 2021 (1am AEST, 3am NZDT)
Thursday, April 8, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 9, 2021 (1am AEST, 3am NZDT)
Abstract: There are several methods to obtain a lower bound for the mean value of the absolute value of the remainder term of the prime number formula as function of a hypothetical zero of the Riemann Zeta function off the critical line. (The case when the Riemann Hypothesis is true can be treated easier.) The most efficient ones include results of Knapowski-Turán, Sz. Gy. Révész , and the author, proved by several different methods.
Abstract: There are several methods to obtain a lower bound for the mean value of the absolute value of the remainder term of the prime number formula as function of a hypothetical zero of the Riemann Zeta function off the critical line. (The case when the Riemann Hypothesis is true can be treated easier.) The most efficient ones include results of Knapowski-Turán, Sz. Gy. Révész , and the author, proved by several different methods.
The result to be proved in the lecture provides (again with an other method) a quite good lower bound and it has the good feature (which is useful in further applications too) that instead of the whole interval [0,X] it gives a good lower bound for the average on [F(X), X] with log F(X) close to log X (that is on "short" intervals measured with the logarithmic scale).
The result to be proved in the lecture provides (again with an other method) a quite good lower bound and it has the good feature (which is useful in further applications too) that instead of the whole interval [0,X] it gives a good lower bound for the average on [F(X), X] with log F(X) close to log X (that is on "short" intervals measured with the logarithmic scale).
Boris Adamczewski, Furstenberg's conjecture, Mahler's method, and finite automata
(CNRS, Université Claude Bernard Lyon 1)
Boris Adamczewski, Furstenberg's conjecture, Mahler's method, and finite automata
(CNRS, Université Claude Bernard Lyon 1)
Thursday, April 1, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 2, 2021 (2am AEDT, 4am NZDT)
Thursday, April 1, 2021 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 2, 2021 (2am AEDT, 4am NZDT)
Abstract: It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.
Abstract: It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.
Vitaly Bergelson, A "soft" dynamical approach to the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
(Ohio State University)
Vitaly Bergelson, A "soft" dynamical approach to the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
(Ohio State University)
Thursday, March 25, 2021 (1:30pm PDT, 4:30pm EDT, 8:30pm GMT, 9:30pm CET, 10:30pm Israel Standard Time)
Thursday, March 25, 2021 (1:30pm PDT, 4:30pm EDT, 8:30pm GMT, 9:30pm CET, 10:30pm Israel Standard Time)
Friday, March 26, 2021 (2am Indian Standard Time, 4:30am CST, 7:30am AEDT, 9:30am NZDT)
Friday, March 26, 2021 (2am Indian Standard Time, 4:30am CST, 7:30am AEDT, 9:30am NZDT)
Abstract: We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdős-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,×).
Abstract: We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdős-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,×).
The talk is based on recent joint work with Florian Richter.
The talk is based on recent joint work with Florian Richter.
Shabnam Akhtari, Orders in Quartic Number Fields and Classical Diophantine Equations
(University of Oregon)
Shabnam Akhtari, Orders in Quartic Number Fields and Classical Diophantine Equations
(University of Oregon)
Thursday, March 18, 2021 (1pm PDT, 4pm EDT, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Thursday, March 18, 2021 (1pm PDT, 4pm EDT, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Friday, March 19, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Friday, March 19, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Chantal David, Moments and non-vanishing of cubic Dirichlet L-functions at s=1/2
(Concordia University)
Chantal David, Moments and non-vanishing of cubic Dirichlet L-functions at s=1/2
(Concordia University)
Thursday, March 11, 2021 (12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Thursday, March 11, 2021 (12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Friday, March 12, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Friday, March 12, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Sanju Velani, The Shrinking Target Problem for Matrix Transformations of Tori
(University of York)
Sanju Velani, The Shrinking Target Problem for Matrix Transformations of Tori
(University of York)
Thursday, March 4, 2021 (12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Thursday, March 4, 2021 (12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time)
Friday, March 5, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Friday, March 5, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)
Ben Green, New lower bounds for van der Waerden numbers
(University of Oxford)
Ben Green, New lower bounds for van der Waerden numbers
(University of Oxford)
Thursday, February 25, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, February 25, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 26, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, February 26, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Colour {1,..,N} red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about sqrt{N}. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about exp ((log N)^{3/4}), and in particular less than any fixed power of N.
Abstract: Colour {1,..,N} red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about sqrt{N}. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about exp ((log N)^{3/4}), and in particular less than any fixed power of N.
I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory.
I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory.
Gabriel Dill, Unlikely Intersections and Distinguished Categories
(University of Oxford)
Gabriel Dill, Unlikely Intersections and Distinguished Categories
(University of Oxford)
Thursday, February 18, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, February 18, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 19, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, February 19, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: After a general introduction to the field of unlikely intersections, I present current work in progress with Fabrizio Barroero, in which we propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows to us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This also yields new unconditional results on the Zilber-Pink conjecture for curves in various contexts.
Abstract: After a general introduction to the field of unlikely intersections, I present current work in progress with Fabrizio Barroero, in which we propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows to us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This also yields new unconditional results on the Zilber-Pink conjecture for curves in various contexts.
Don Zagier, Analytic functions related to zeta-values, cotangent products, and the cohomology of SL_2(\Z)
(Max Planck Institute for Mathematics)
Don Zagier, Analytic functions related to zeta-values, cotangent products, and the cohomology of SL_2(\Z)
(Max Planck Institute for Mathematics)
Thursday, February 11, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, February 11, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 12, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, February 12, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: I will report on the properties of various functions, going back essentially to Herglotz, that relate to a number of different topics in number theory, including those in the title but also others like Hecke operators or Stark's conjectures. This is joint work with Danylo Radchenko.
Abstract: I will report on the properties of various functions, going back essentially to Herglotz, that relate to a number of different topics in number theory, including those in the title but also others like Hecke operators or Stark's conjectures. This is joint work with Danylo Radchenko.
Oleksiy Klurman, On the zeros of Fekete polynomials
(University of Bristol)
Oleksiy Klurman, On the zeros of Fekete polynomials
(University of Bristol)
Thursday, February 4, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, February 4, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 5, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, February 5, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem. The goal of this talk is to discuss what we knew, know and would like to know about zeros of such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.
Abstract: Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem. The goal of this talk is to discuss what we knew, know and would like to know about zeros of such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.
William Banks, On the distribution of reduced fractions with squarefree denominators
(University of Missouri)
William Banks, On the distribution of reduced fractions with squarefree denominators
(University of Missouri)
Thursday, January 28, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, January 28, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 29, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, January 29, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In this talk we discuss how the nonvanishing of the Riemann zeta function in a half-plane {sigma>sigma_0}, with some real sigma_0<1, is equivalent to a strong statement about the distribution in the unit interval of reduced fractions with squarefree denominators.
Abstract: In this talk we discuss how the nonvanishing of the Riemann zeta function in a half-plane {sigma>sigma_0}, with some real sigma_0<1, is equivalent to a strong statement about the distribution in the unit interval of reduced fractions with squarefree denominators.
The approach utilizes an unconditional generalization of a theorem of Blomer concerning the distribution "on average" of squarefree integers in arithmetic progressions to large moduli.
The approach utilizes an unconditional generalization of a theorem of Blomer concerning the distribution "on average" of squarefree integers in arithmetic progressions to large moduli.
Lior Bary-Soroker, Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic
(Tel Aviv University)
Lior Bary-Soroker, Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic
(Tel Aviv University)
Thursday, January 21, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, January 21, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 22, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, January 22, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: In the talk we will discuss recent advances on the following two questions:
Abstract: In the talk we will discuss recent advances on the following two questions:
Let A(X) = sum ±X^i be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.
Let A(X) = sum ±X^i be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.
Q1: What is the probability that A is irreducible as the degree goes to infinity?
Q1: What is the probability that A is irreducible as the degree goes to infinity?
Q2: What is the typical Galois group of A?
Q2: What is the typical Galois group of A?
One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.
One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.
Peter Sarnak, Summation formulae in spectral theory and number theory
(Institute for Advanced Study and Princeton University)
Peter Sarnak, Summation formulae in spectral theory and number theory
(Institute for Advanced Study and Princeton University)
A talk in honor of Zeev Rudnick's 60th birthday
Special Chair: Lior Bary-Soroker
Special Chair: Lior Bary-Soroker
Thursday, January 14, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Thursday, January 14, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 15, 2021 (12am CST, 3am AEDT, 5am NZDT)
Friday, January 15, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: The Poisson Summation formula, Riemann-Guinand-Weil explicit formula, Selberg Trace Formula and Lefschetz Trace formula in the function field, are starting points for a number of Zeev Rudnick's works. We will review some of these before describing some recent applications (joint with P. Kurasov) of Lang's G_m conjectures to the additive structure of the spectra of metric graphs and crystalline measures.
Abstract: The Poisson Summation formula, Riemann-Guinand-Weil explicit formula, Selberg Trace Formula and Lefschetz Trace formula in the function field, are starting points for a number of Zeev Rudnick's works. We will review some of these before describing some recent applications (joint with P. Kurasov) of Lang's G_m conjectures to the additive structure of the spectra of metric graphs and crystalline measures.
Imre Ruzsa, Additive decomposition of signed primes
(Alfréd Rényi Institute of Mathematics)
Imre Ruzsa, Additive decomposition of signed primes
(Alfréd Rényi Institute of Mathematics)
Thursday, January 7, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, January 8, 2021 (12am CST, 3am AEDT, 5am NZDT)
Thursday, January 7, 2021 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, January 8, 2021 (12am CST, 3am AEDT, 5am NZDT)
Abstract: Assuming the prime-tuple hypothesis, the set of signed primes is a sumset. More exactly, there are infinite sets A, B of integers such that A+B consists exactly of the (positive and negative) primes with |p|>3. I will also meditate on the possibility of a triple sum and analogous problems for the set of squarefree numbers.
Abstract: Assuming the prime-tuple hypothesis, the set of signed primes is a sumset. More exactly, there are infinite sets A, B of integers such that A+B consists exactly of the (positive and negative) primes with |p|>3. I will also meditate on the possibility of a triple sum and analogous problems for the set of squarefree numbers.
Jianya Liu, Mobius disjointness for irregular flows
(Shandong University)
Jianya Liu, Mobius disjointness for irregular flows
(Shandong University)
Tuesday, December 22, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, December 22, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Abstract: The behavior of the Mobius function is central in the theory of prime numbers. A surprising connection with the theory of dynamical systems was discovered in 2010 by P. Sarnak, who formulated the Mobius Disjointness Conjecture (MDC), which asserts that the Mobius function is linearly disjoint from any zero-entropy flows. This conjecture opened the way into a large body of research on the interface of analytic number theory and ergodic theory. In this talk I will report how to establish MDC for a class of irregular flows, which are in general mysterious and ill understood. This is based on joint works with P. Sarnak, and with W. Huang and K. Wang.
Abstract: The behavior of the Mobius function is central in the theory of prime numbers. A surprising connection with the theory of dynamical systems was discovered in 2010 by P. Sarnak, who formulated the Mobius Disjointness Conjecture (MDC), which asserts that the Mobius function is linearly disjoint from any zero-entropy flows. This conjecture opened the way into a large body of research on the interface of analytic number theory and ergodic theory. In this talk I will report how to establish MDC for a class of irregular flows, which are in general mysterious and ill understood. This is based on joint works with P. Sarnak, and with W. Huang and K. Wang.
Gisbert Wüstholz, Baker's theory for 1-motives
(ETH / University Zurich)
Gisbert Wüstholz, Baker's theory for 1-motives
(ETH / University Zurich)
Thursday, December 17, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 18, 2020 (12am CST, 3am AEDT, 5am NZDT)
Thursday, December 17, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 18, 2020 (12am CST, 3am AEDT, 5am NZDT)
Abstract: From a historical point of view transcendence theory used to be a nice collection of mostly particular results, very difficult to find and to prove. To find numbers for which one has a chance to prove transcendence is very difficult. To state conjecture is not so difficult but in most cases hopeless to prove. In our lecture we try to draw a picture of quite far reaching frames in the theory of motives which can put transcendence theory into a more conceptual setting.
Abstract: From a historical point of view transcendence theory used to be a nice collection of mostly particular results, very difficult to find and to prove. To find numbers for which one has a chance to prove transcendence is very difficult. To state conjecture is not so difficult but in most cases hopeless to prove. In our lecture we try to draw a picture of quite far reaching frames in the theory of motives which can put transcendence theory into a more conceptual setting.
Looking at periods of rational 1-forms on varieties we realized that there is a more conceptual background behind the properties of these complex numbers than had been thought so far. The central question which I was trying for more than three decades to answer was to determine when a period is algebraic. A priori a period is zero, algebraic or transcendental, no surprise! It is also not difficult to give examples for cases when periods are algebraic. However the big question was whether the examples are all examples. Quite recently, partly jointly with Annette Huber we developed a new transcendence theory within 1-motives which extend commutative algebraic groups. One outcome was that algebraicity of periods has a very conceptual description and we shall give a precise and surprisingly simple answer.
Looking at periods of rational 1-forms on varieties we realized that there is a more conceptual background behind the properties of these complex numbers than had been thought so far. The central question which I was trying for more than three decades to answer was to determine when a period is algebraic. A priori a period is zero, algebraic or transcendental, no surprise! It is also not difficult to give examples for cases when periods are algebraic. However the big question was whether the examples are all examples. Quite recently, partly jointly with Annette Huber we developed a new transcendence theory within 1-motives which extend commutative algebraic groups. One outcome was that algebraicity of periods has a very conceptual description and we shall give a precise and surprisingly simple answer.
Many questions which were central in transcendence theory and with a long and famous history turn out to get a general answer within the new theory. The classical work of Baker turns out to be a very special but seminal case.
Many questions which were central in transcendence theory and with a long and famous history turn out to get a general answer within the new theory. The classical work of Baker turns out to be a very special but seminal case.
Adam Harper, Large fluctuations of random multiplicative functions
(University of Warwick)
Adam Harper, Large fluctuations of random multiplicative functions
(University of Warwick)
Tuesday, December 15, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, December 15, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Abstract: Random multiplicative functions $f(n)$ are a well studied random model for deterministic multiplicative functions like Dirichlet characters or the Mobius function. Arguably the first question ever studied about them, by Wintner in 1944, was to obtain almost sure bounds for the largest fluctuations of their partial $\sum_{n \leq x} f(n)$, seeking to emulate the classical Law of the Iterated Logarithm for independent random variables. It remains an open question to sharply determine the size of these fluctuations, and in this talk I will describe a new result in that direction. I hope to get to some interesting details of the new proof in the latter part of the talk, but most of the discussion should be widely accessible.
Abstract: Random multiplicative functions $f(n)$ are a well studied random model for deterministic multiplicative functions like Dirichlet characters or the Mobius function. Arguably the first question ever studied about them, by Wintner in 1944, was to obtain almost sure bounds for the largest fluctuations of their partial $\sum_{n \leq x} f(n)$, seeking to emulate the classical Law of the Iterated Logarithm for independent random variables. It remains an open question to sharply determine the size of these fluctuations, and in this talk I will describe a new result in that direction. I hope to get to some interesting details of the new proof in the latter part of the talk, but most of the discussion should be widely accessible.
Maksym Radziwill, The Fyodorov-Hiary-Keating conjecture
(California Institute of Technology)
Maksym Radziwill, The Fyodorov-Hiary-Keating conjecture
(California Institute of Technology)
Thursday, December 10, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 11, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, December 10, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 11, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: I will discuss recent progress on the Fyodorov-Hiary-Keating conjecture on the distribution of the local maximum of the Riemann zeta-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.
Abstract: I will discuss recent progress on the Fyodorov-Hiary-Keating conjecture on the distribution of the local maximum of the Riemann zeta-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.
Jacob Tsimerman, Bounding torsion in class group and families of local systems (University of Toronto)
Jacob Tsimerman, Bounding torsion in class group and families of local systems (University of Toronto)
Please note the unusual time!
Please note the unusual time!
Monday, December 7, 2020 (2pm PST, 5pm EST, 10pm GMT, 11pm CET)
Tuesday, December 8, 2020 (12am Israel Standard Time, 3:30am Indian Standard Time, 6am CST, 9am AEDT, 11am NZDT)
Monday, December 7, 2020 (2pm PST, 5pm EST, 10pm GMT, 11pm CET)
Tuesday, December 8, 2020 (12am Israel Standard Time, 3:30am Indian Standard Time, 6am CST, 9am AEDT, 11am NZDT)
Abstract: (joint w/ Arul Shankar) We discuss a new method to bound 5-torsion in class groups of quadratic fields using the refined BSD conjecture for elliptic curves. The most natural “trivial” bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a selmer group of a Galois module. We may then similarly bound its size by the Tate-Shafarevich group of an appropriate elliptic curve, which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds selmer groups of finite Galois modules by embedding into global objects, and using class number formulas. If time permits, we explain how the function field picture yields unconditional results and suggests further generalizations.
Abstract: (joint w/ Arul Shankar) We discuss a new method to bound 5-torsion in class groups of quadratic fields using the refined BSD conjecture for elliptic curves. The most natural “trivial” bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a selmer group of a Galois module. We may then similarly bound its size by the Tate-Shafarevich group of an appropriate elliptic curve, which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds selmer groups of finite Galois modules by embedding into global objects, and using class number formulas. If time permits, we explain how the function field picture yields unconditional results and suggests further generalizations.
Alexander Lubotzky, From Ramanujan graphs to Ramanujan complexes
(Hebrew University of Jerusalem)
Alexander Lubotzky, From Ramanujan graphs to Ramanujan complexes
(Hebrew University of Jerusalem)
Thursday, December 3, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 4, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, December 3, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, December 4, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: Ramanujan graphs are k-regular graphs with all non trivial eigenvalues bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups. The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms.
Abstract: Ramanujan graphs are k-regular graphs with all non trivial eigenvalues bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups. The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms.
The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplicial complexes which on one hand are "random like" and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.
The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplicial complexes which on one hand are "random like" and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.
Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will survey some of these applications.
Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will survey some of these applications.
Dragos Ghioca, A couple of conjectures in arithmetic dynamics over fields of positive characteristic
(University of British Columbia)
Dragos Ghioca, A couple of conjectures in arithmetic dynamics over fields of positive characteristic
(University of British Columbia)
Monday, November 30, 2020 (5pm PST, 8pm EST)
Tuesday, December 1, 2020 (1am GMT, 2am CET, 3am Israel Standard Time, 6:30am IST, 9am China Standard Time, 12pm AEDT, 2pm NZDT)
Monday, November 30, 2020 (5pm PST, 8pm EST)
Tuesday, December 1, 2020 (1am GMT, 2am CET, 3am Israel Standard Time, 6:30am IST, 9am China Standard Time, 12pm AEDT, 2pm NZDT)
Abstract: The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.
Abstract: The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.
Michael Stoll, An application of "Selmer group Chabauty" to arithmetic dynamics
(University of Bayreuth)
Michael Stoll, An application of "Selmer group Chabauty" to arithmetic dynamics
(University of Bayreuth)
Thursday, November 26, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 27, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, November 26, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 27, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: The irreducibility or otherwise of iterates of polynomials is an important question in arithmetic dynamics. For example, it is conjectured that whenever the second iterate of x^2 + c (with c a rational number) is irreducible over Q, then so are all iterates. A sufficient criterion for the iterates to be irreducible can be expressed in terms of rational points on certain hyperelliptic curves. We will show how to use the "Selmer group Chabauty" method developed by the speaker to determine the set of rational points on a hyperelliptic curve of genus 7. This leads to a proof that the seventh iterate of x^2 + c must be irreducible if the second iterate is. Assuming GRH, we can extend this to the tenth iterate.
Abstract: The irreducibility or otherwise of iterates of polynomials is an important question in arithmetic dynamics. For example, it is conjectured that whenever the second iterate of x^2 + c (with c a rational number) is irreducible over Q, then so are all iterates. A sufficient criterion for the iterates to be irreducible can be expressed in terms of rational points on certain hyperelliptic curves. We will show how to use the "Selmer group Chabauty" method developed by the speaker to determine the set of rational points on a hyperelliptic curve of genus 7. This leads to a proof that the seventh iterate of x^2 + c must be irreducible if the second iterate is. Assuming GRH, we can extend this to the tenth iterate.
Jasmin Matz, Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect
(University of Copenhagen)
Jasmin Matz, Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect
(University of Copenhagen)
Tuesday, November 24, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, November 24, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Abstract: Suppose M is a closed Riemannian manifold with an orthonormal basis B of L^2(M) consisting of Laplace eigenfunctions. A classical result of Shnirelman and others proves that if the geodesic flow on the cotangent bundle of M is ergodic, then M is quantum ergodic, in particular, on average, the probability measures defined by the functions f in B on M tends on average towards the Riemannian measure on M in the high energy limit (i.e, as the Laplace eigenvalues of f -> infinity).
We now want to look at a level aspect of this property, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds M_j together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of SL(n,R)/SO(n).
Abstract: Suppose M is a closed Riemannian manifold with an orthonormal basis B of L^2(M) consisting of Laplace eigenfunctions. A classical result of Shnirelman and others proves that if the geodesic flow on the cotangent bundle of M is ergodic, then M is quantum ergodic, in particular, on average, the probability measures defined by the functions f in B on M tends on average towards the Riemannian measure on M in the high energy limit (i.e, as the Laplace eigenvalues of f -> infinity).
We now want to look at a level aspect of this property, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds M_j together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of SL(n,R)/SO(n).
Jason Bell, A transcendental dynamical degree
(University of Waterloo)
Jason Bell, A transcendental dynamical degree
(University of Waterloo)
Monday, November 16, 2020 (5pm PST, 8pm EST)
Tuesday, November 17, 2020 (1am GMT, 2am CET, 3am Israel Standard Time, 6:30am IST, 9am China Standard Time, 12pm AEDT, 2pm NZDT)
Monday, November 16, 2020 (5pm PST, 8pm EST)
Tuesday, November 17, 2020 (1am GMT, 2am CET, 3am Israel Standard Time, 6:30am IST, 9am China Standard Time, 12pm AEDT, 2pm NZDT)
Abstract: The degree of a dominant rational map $f:\mathbb{P}^n\to \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\deg(f^n)$, which is submultiplicative and hence has the property that there is some $\lambda\ge 1$ such that $(\deg(f^n))^{1/n}\to \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We’ll give an overview of the significance of the dynamical degree in complex dynamics and describe an example in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller and Mattias Jonsson.
Abstract: The degree of a dominant rational map $f:\mathbb{P}^n\to \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\deg(f^n)$, which is submultiplicative and hence has the property that there is some $\lambda\ge 1$ such that $(\deg(f^n))^{1/n}\to \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We’ll give an overview of the significance of the dynamical degree in complex dynamics and describe an example in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller and Mattias Jonsson.
David Masser, Pencils of norm form equations and a conjecture of Thomas
(University of Basel)
David Masser, Pencils of norm form equations and a conjecture of Thomas
(University of Basel)
Thursday, November 12, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 13, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, November 12, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 13, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: We consider certain one-parameter families of norm form (and other) diophantine equations, and we solve them completely and uniformly for all sufficiently large positive integer values of the parameter (everything effective), following a line started by Emery Thomas in 1990. The new tool is a bounded height result from 2017 by Francesco Amoroso, Umberto Zannier and the speaker.
Abstract: We consider certain one-parameter families of norm form (and other) diophantine equations, and we solve them completely and uniformly for all sufficiently large positive integer values of the parameter (everything effective), following a line started by Emery Thomas in 1990. The new tool is a bounded height result from 2017 by Francesco Amoroso, Umberto Zannier and the speaker.
Gérald Tenenbaum, Recent progress on the Selberg-Delange method in analytic number theory
(Université de Lorraine)
Gérald Tenenbaum, Recent progress on the Selberg-Delange method in analytic number theory
(Université de Lorraine)
Tuesday, November 10, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, November 10, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Pär Kurlberg, Distribution of lattice points on hyperbolic circles
(KTH)
Pär Kurlberg, Distribution of lattice points on hyperbolic circles
(KTH)
Thursday, November 5, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 6, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, November 5, 2020 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, November 6, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: We study the distribution of lattice points lying on expanding circles in the hyperbolic plane. The angles of lattice points arising from the orbit of the modular group PSL(2,Z), and lying on hyperbolic circles centered at i, are shown to be equidistributed for generic radii (among the ones that contain points). We also show that angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of euclidean lattice points lying on circles in the plane, along a thin subsequence of radii. This is joint work with D. Chatzakos, S. Lester and I. Wigman.
Abstract: We study the distribution of lattice points lying on expanding circles in the hyperbolic plane. The angles of lattice points arising from the orbit of the modular group PSL(2,Z), and lying on hyperbolic circles centered at i, are shown to be equidistributed for generic radii (among the ones that contain points). We also show that angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of euclidean lattice points lying on circles in the plane, along a thin subsequence of radii. This is joint work with D. Chatzakos, S. Lester and I. Wigman.
Jens Marklof, The three gap theorem in higher dimensions
(University of Bristol)
Jens Marklof, The three gap theorem in higher dimensions
(University of Bristol)
Tuesday, November 3, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, November 3, 2020 (2am PST, 5am EST, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Abstract: Take a point on the unit circle and rotate it N times by a fixed angle. The N points thus generated partition the circle into N intervals. A beautiful fact, first conjectured by Hugo Steinhaus in the 1950s and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski, is that for any choice of N, no matter how large, these intervals can have at most three distinct lengths. In this lecture I will explore an interpretation of the three gap theorem in terms of the space of Euclidean lattices, which will produce various new results in higher dimensions, including gaps in the fractional parts of linear forms and nearest neighbour distances in multi-dimensional Kronecker sequences. The lecture is based on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).
Abstract: Take a point on the unit circle and rotate it N times by a fixed angle. The N points thus generated partition the circle into N intervals. A beautiful fact, first conjectured by Hugo Steinhaus in the 1950s and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski, is that for any choice of N, no matter how large, these intervals can have at most three distinct lengths. In this lecture I will explore an interpretation of the three gap theorem in terms of the space of Euclidean lattices, which will produce various new results in higher dimensions, including gaps in the fractional parts of linear forms and nearest neighbour distances in multi-dimensional Kronecker sequences. The lecture is based on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).
- J. Marklof and A. Strömbergsson, The three gap theorem and the space of lattices, American Mathematical Monthly 124 (2017) 741-745 https://people.maths.bris.ac.uk/~majm/bib/threegap.pdf
- A. Haynes and J. Marklof, Higher dimensional Steinhaus and Slater problems via homogeneous dynamics, Annales scientifiques de l'Ecole normale superieure 53 (2020) 537-557 https://people.maths.bris.ac.uk/~majm/bib/steinhaus.pdf
- A. Haynes and J. Marklof, A five distance theorem for Kronecker sequences, preprint arXiv:2009.08444 https://people.maths.bris.ac.uk/~majm/bib/steinhaus2.pdf
Will Sawin, The distribution of prime polynomials over finite fields
(Columbia University)
Will Sawin, The distribution of prime polynomials over finite fields
(Columbia University)
Thursday, October 29, 2020 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, October 30, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Thursday, October 29, 2020 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm IST)
Friday, October 30, 2020 (12am China Standard Time, 3am AEDT, 5am NZDT)
Abstract: Many conjectures in number theory have analogues for polynomials in one variable over a finite field. In recent works with Mark Shusterman, we proved analogues of two conjectures about prime numbers - the twin primes conjecture and the conjecture that there are infinitely many primes of the form n²+1. I will describe these results and explain some of the key ideas in the proofs, which combine classical analytic methods, elementary algebraic manipulations, and geometric methods.
Abstract: Many conjectures in number theory have analogues for polynomials in one variable over a finite field. In recent works with Mark Shusterman, we proved analogues of two conjectures about prime numbers - the twin primes conjecture and the conjecture that there are infinitely many primes of the form n²+1. I will describe these results and explain some of the key ideas in the proofs, which combine classical analytic methods, elementary algebraic manipulations, and geometric methods.
Gal Binyamini, Point counting for foliations in Diophantine geometry
(Weizmann Institute of Science)
Gal Binyamini, Point counting for foliations in Diophantine geometry
(Weizmann Institute of Science)
Tuesday, October 27, 2020 (3am PDT, 6am EDT, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Tuesday, October 27, 2020 (3am PDT, 6am EDT, 10am GMT, 11am CET, 12pm Israel Standard Time, 3:30pm IST, 6pm China Standard Time, 9pm AEDT, 11pm NZDT)
Abstract: I will discuss "point counting" in two broad senses: counting the intersections between a trascendental variety and an algebraic one; and counting the number of algebraic points, as a function of degree and height, on a transcendental variety. After reviewing the fundamental results in this area - from the theory of o-minimal structures and the Pila-Wilkie theorem, I will restrict attention to the case that the transcendental variety is given in terms of a leaf of an algebraic foliation, and everything is defined over a number field. It turns out that in this case far stronger estimates can be obtained.
Abstract: I will discuss "point counting" in two broad senses: counting the intersections between a trascendental variety and an algebraic one; and counting the number of algebraic points, as a function of degree and height, on a transcendental variety. After reviewing the fundamental results in this area - from the theory of o-minimal structures and the Pila-Wilkie theorem, I will restrict attention to the case that the transcendental variety is given in terms of a leaf of an algebraic foliation, and everything is defined over a number field. It turns out that in this case far stronger estimates can be obtained.
Applying the above to foliations associated to principal G-bundles on various moduli spaces, many classical application of the Pila-Wilkie theorem can be sharpened and effectivized. In particular I will discuss issues around effectivity and polynomial-time solvability for the Andre-Oort conjecture, unlikely intersections in abelian schemes, and some related directions.
Applying the above to foliations associated to principal G-bundles on various moduli spaces, many classical application of the Pila-Wilkie theorem can be sharpened and effectivized. In particular I will discuss issues around effectivity and polynomial-time solvability for the Andre-Oort conjecture, unlikely intersections in abelian schemes, and some related directions.
Sergei Konyagin, A construction of A. Schinzel - many numbers in a short interval without small prime factors
(Steklov Institute of Mathematics)
Sergei Konyagin, A construction of A. Schinzel - many numbers in a short interval without small prime factors
(Steklov Institute of Mathematics)
Thursday, October 22, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, October 23, 2020 (2am AEDT, 4am NZDT)
Thursday, October 22, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, October 23, 2020 (2am AEDT, 4am NZDT)
Jörg Brüdern, Harmonic analysis of arithmetic functions
(University of Göttingen)
Jörg Brüdern, Harmonic analysis of arithmetic functions
(University of Göttingen)
Tuesday, October 20, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 8pm AEDT, 10pm NZDT)
Tuesday, October 20, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 8pm AEDT, 10pm NZDT)
Abstract: We study arithmetic functions that are bounded in mean square, and simultaneously have a mean value over any arithmetic progression. A Besicovitch type norm makes the set of these functions a Banach space. We apply the Hardy-Littlewood (circle) method to analyse this space. This method turns out to be a surprisingly flexible tool for this purpose. We obtain several characterisations of limit periodic functions, correlation formulae, and we give some applications to Waring's problem and related topics. Finally, we direct the theory to the distribution of the arithmetic functions under review in arithmetic progressions, with mean square results of Barban-Davenport-Halberstam type and related asymptotic formulae at the focus of our attention. There is a rich literature on this last theme. Our approach supersedes previous work in various ways, and ultimately provides another characterisation of limit periodic functions: the variance over arithmetic progression is atypically small if and only if the input function is limit periodic.
Abstract: We study arithmetic functions that are bounded in mean square, and simultaneously have a mean value over any arithmetic progression. A Besicovitch type norm makes the set of these functions a Banach space. We apply the Hardy-Littlewood (circle) method to analyse this space. This method turns out to be a surprisingly flexible tool for this purpose. We obtain several characterisations of limit periodic functions, correlation formulae, and we give some applications to Waring's problem and related topics. Finally, we direct the theory to the distribution of the arithmetic functions under review in arithmetic progressions, with mean square results of Barban-Davenport-Halberstam type and related asymptotic formulae at the focus of our attention. There is a rich literature on this last theme. Our approach supersedes previous work in various ways, and ultimately provides another characterisation of limit periodic functions: the variance over arithmetic progression is atypically small if and only if the input function is limit periodic.
Cameron L. Stewart, On integers represented by binary forms
(University of Waterloo)
Cameron L. Stewart, On integers represented by binary forms
(University of Waterloo)
Thursday, October 15, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 16, 2020 (2am AEDT, 4am NZDT)
Thursday, October 15, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 16, 2020 (2am AEDT, 4am NZDT)
Abstract: We shall discuss the following results which are joint work with Stanley Xiao.
Abstract: We shall discuss the following results which are joint work with Stanley Xiao.
Let F(x,y) be a binary form with integer coefficients, degree d(>2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z^(2/d).
Let F(x,y) be a binary form with integer coefficients, degree d(>2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z^(2/d).
Let k be an integer with k>1 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z^(2/d).
Let k be an integer with k>1 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z^(2/d).
Alexander Gorodnik, Arithmetic approach to the spectral gap problem
(University of Zurich)
Alexander Gorodnik, Arithmetic approach to the spectral gap problem
(University of Zurich)
Tuesday, October 13, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 8pm AEDT, 10pm NZDT)
Tuesday, October 13, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 8pm AEDT, 10pm NZDT)
Abstract: The spectral gap is an analytic property of group actions which can be described as absence of "almost invariant vectors" or more quantitatively in terms of norm bounds for suitable averaging operators. In the setting of homogeneous spaces this property also has a profound number-theoretic meaning since it is closely related to understanding the automorphic representations. In this talk we survey some previous results about the spectral gap property and describe new approaches to deriving upper and lower bounds for the spectral gap.
Abstract: The spectral gap is an analytic property of group actions which can be described as absence of "almost invariant vectors" or more quantitatively in terms of norm bounds for suitable averaging operators. In the setting of homogeneous spaces this property also has a profound number-theoretic meaning since it is closely related to understanding the automorphic representations. In this talk we survey some previous results about the spectral gap property and describe new approaches to deriving upper and lower bounds for the spectral gap.
Philippe Michel, Simultaneous reductions of CM elliptic curves
(EPFL)
Philippe Michel, Simultaneous reductions of CM elliptic curves
(EPFL)
Thursday, October 8, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 9, 2020 (2am AEDT, 4am NZDT)
Thursday, October 8, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 9, 2020 (2am AEDT, 4am NZDT)
Abstract: Let $E$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of discriminant $D<0$. Given $p$ a prime ; if $p$ is inert or ramified in the quadratic field generated by $\sqrt D$ then $E$ has supersingular reduction at a(ny) fixed place above $p$. By a variant of Duke’s equidistribution theorem, as $D$ grows along such discriminants, the proportion of CM elliptic curves with CM by $O_D$ whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes $p_1,\cdots,p_s$ and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank $2$ actions on products of locally homogeneous spaces implies that as $D$ grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual $p_i$ (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement -- obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.
Abstract: Let $E$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of discriminant $D<0$. Given $p$ a prime ; if $p$ is inert or ramified in the quadratic field generated by $\sqrt D$ then $E$ has supersingular reduction at a(ny) fixed place above $p$. By a variant of Duke’s equidistribution theorem, as $D$ grows along such discriminants, the proportion of CM elliptic curves with CM by $O_D$ whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes $p_1,\cdots,p_s$ and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank $2$ actions on products of locally homogeneous spaces implies that as $D$ grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual $p_i$ (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement -- obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.
Alexander Smith, Selmer groups and a Cassels-Tate pairing for finite Galois modules
(Harvard University)
Alexander Smith, Selmer groups and a Cassels-Tate pairing for finite Galois modules
(Harvard University)
Monday, October 5, 2020 (5pm PDT, 8pm EDT)
Tuesday, October 6, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 11am AEDT, 1pm NZDT)
Monday, October 5, 2020 (5pm PDT, 8pm EDT)
Tuesday, October 6, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 11am AEDT, 1pm NZDT)
Abstract: I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.
Abstract: I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.
As one application of this theory, I will prove an elementary result on the symmetry of the class group pairing for number fields with many roots of unity and connect this to the work of mine and others on class group statistics.
As one application of this theory, I will prove an elementary result on the symmetry of the class group pairing for number fields with many roots of unity and connect this to the work of mine and others on class group statistics.
This work is joint with Adam Morgan.
This work is joint with Adam Morgan.
Wei Ho, The Hasse local-to-global principle for some genus one curves
(University of Michigan)
Wei Ho, The Hasse local-to-global principle for some genus one curves
(University of Michigan)
Thursday, October 1, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 2, 2020 (1am AEST, 4am NZDT)
Thursday, October 1, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 2, 2020 (1am AEST, 4am NZDT)
Abstract: The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails.
Abstract: The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails.
In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, with a special focus on genus one curves given by bihomogeneous polynomials of bidegree (2,2) in \mathbb{P}^1 \times \mathbb{P}^1. For example, we will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves, by comparing the average sizes of 2- and 3-Selmer groups for a family of elliptic curves with a marked point (joint work with Manjul Bhargava).
In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, with a special focus on genus one curves given by bihomogeneous polynomials of bidegree (2,2) in \mathbb{P}^1 \times \mathbb{P}^1. For example, we will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves, by comparing the average sizes of 2- and 3-Selmer groups for a family of elliptic curves with a marked point (joint work with Manjul Bhargava).
Julie Tzu-Yueh Wang, Pisot's d-th root's conjecture for function fields and its complex analog
(Academia Sinica, Taiwan)
Julie Tzu-Yueh Wang, Pisot's d-th root's conjecture for function fields and its complex analog
(Academia Sinica, Taiwan)
Monday, September 28, 2020 (5pm PDT, 8pm EDT)
Tuesday, September 29, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 1pm NZDT)
Monday, September 28, 2020 (5pm PDT, 8pm EDT)
Tuesday, September 29, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 1pm NZDT)
Emmanuel Breuillard, A subspace theorem for manifolds
(University of Cambridge)
Emmanuel Breuillard, A subspace theorem for manifolds
(University of Cambridge)
Thursday, September 24, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, September 25, 2020 (1am AEST, 3am NZST)
Thursday, September 24, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, September 25, 2020 (1am AEST, 3am NZST)
Abstract: In the late 90's Kleinbock and Margulis solved a long-standing conjecture due to Sprindzuk regarding diophantine approximation on submanifolds of R^n. Their method used homogeneous dynamics via the so-called non-divergence estimates for unipotent flows on the space of lattices. In this talk I will explain how these ideas, combined with a certain understanding of the geometry at the heart of Schmidt's subspace theorem, in particular the notion of Harder-Narasimhan filtration, leads to a metric version of the subspace theorem, where the linear forms are allowed to depend on a parameter. This subspace theorem for manifolds allows to quickly compute certain diophantine exponents, and it leads to several generalizations of the Kleinbock-Margulis results in a variety of contexts. Joint work with Nicolas de Saxcé.
Abstract: In the late 90's Kleinbock and Margulis solved a long-standing conjecture due to Sprindzuk regarding diophantine approximation on submanifolds of R^n. Their method used homogeneous dynamics via the so-called non-divergence estimates for unipotent flows on the space of lattices. In this talk I will explain how these ideas, combined with a certain understanding of the geometry at the heart of Schmidt's subspace theorem, in particular the notion of Harder-Narasimhan filtration, leads to a metric version of the subspace theorem, where the linear forms are allowed to depend on a parameter. This subspace theorem for manifolds allows to quickly compute certain diophantine exponents, and it leads to several generalizations of the Kleinbock-Margulis results in a variety of contexts. Joint work with Nicolas de Saxcé.
Ilya D. Shkredov, Zaremba's conjecture and growth in groups
(Steklov Mathematical Institute, Moscow)
Ilya D. Shkredov, Zaremba's conjecture and growth in groups
(Steklov Mathematical Institute, Moscow)
Tuesday, September 22, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Tuesday, September 22, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Abstract: Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a<q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.
Abstract: Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a<q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.
Özlem Imamoglu, A class number formula of Hurwitz
(ETH Zürich)
Özlem Imamoglu, A class number formula of Hurwitz
(ETH Zürich)
Thursday, September 17, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 18, 2020 (1am AEST, 3am NZST)
Thursday, September 17, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 18, 2020 (1am AEST, 3am NZST)
Abstract: In a little known paper Hurwitz gave an infinite series representation for the class number of positive definite binary quadratic forms In this talk I will report on joint work with W. Duke and A. Toth where we show how the ideas of Hurwitz can be applied in other settings, in particular to give a formula for the class number of binary cubic forms.
Abstract: In a little known paper Hurwitz gave an infinite series representation for the class number of positive definite binary quadratic forms In this talk I will report on joint work with W. Duke and A. Toth where we show how the ideas of Hurwitz can be applied in other settings, in particular to give a formula for the class number of binary cubic forms.
Dzmitry Badziahin, Approximation by algebraic numbers
(University of Sydney)
Dzmitry Badziahin, Approximation by algebraic numbers
(University of Sydney)
Tuesday, September 15, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Tuesday, September 15, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Abstract: In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number x, by w_n^*(x) we define the supremum of all positive real values w such that the inequality
Abstract: In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number x, by w_n^*(x) we define the supremum of all positive real values w such that the inequality
|x - a| < H(a)^{-w-1}
|x - a| < H(a)^{-w-1}
has infinitely many solutions in algebraic real numbers a of degree at most n. Here H(a) means the naive height of the minimal polynomial in Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w_n^*(x) is at least n for all transcendental x? Apart from partial results for small values of n, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w_n^*(x) \ge n/2+1 - o(1). Until recently, the only improvements to this bound were in terms of O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n). More precisely, we show that w_n^*(x) > n/\sqrt{3}.
has infinitely many solutions in algebraic real numbers a of degree at most n. Here H(a) means the naive height of the minimal polynomial in Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w_n^*(x) is at least n for all transcendental x? Apart from partial results for small values of n, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w_n^*(x) \ge n/2+1 - o(1). Until recently, the only improvements to this bound were in terms of O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n). More precisely, we show that w_n^*(x) > n/\sqrt{3}.
Bianca Viray, Existence of quadratic points on intersections of quadrics
(University of Washington)
Bianca Viray, Existence of quadratic points on intersections of quadrics
(University of Washington)
Thursday, September 10, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 11, 2020 (1am AEST, 3am NZST)
Thursday, September 10, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 11, 2020 (1am AEST, 3am NZST)
Abstract: Springer's theorem and the Amer-Brumer theorem together imply that intersections of two quadrics have a rational point if and only if they have a 0-cycle of degree 1. In this talk, we consider whether this statement can be strengthened in the case when there is no rational point, namely whether 1) the least degree of a 0-cycle can be bounded, and 2) whether there is an effective 0-cycle of this degree. We report on results in this direction, paying particular attention to the case of local and global fields. This is joint work with Brendan Creutz.
Abstract: Springer's theorem and the Amer-Brumer theorem together imply that intersections of two quadrics have a rational point if and only if they have a 0-cycle of degree 1. In this talk, we consider whether this statement can be strengthened in the case when there is no rational point, namely whether 1) the least degree of a 0-cycle can be bounded, and 2) whether there is an effective 0-cycle of this degree. We report on results in this direction, paying particular attention to the case of local and global fields. This is joint work with Brendan Creutz.
Maryna Viazovska, Universal optimality, Fourier interpolation, and modular integrals
(EPFL)
Maryna Viazovska, Universal optimality, Fourier interpolation, and modular integrals
(EPFL)
Tuesday, September 8, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11pm IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, September 8, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11pm IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: In this lecture we will show that the E8 and Leech lattices minimize energy for a wide class of potential functions. This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. Construction of the optimal auxiliary functions attaining these bounds is based on a new interpolation theorem. This is joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko.
Abstract: In this lecture we will show that the E8 and Leech lattices minimize energy for a wide class of potential functions. This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. Construction of the optimal auxiliary functions attaining these bounds is based on a new interpolation theorem. This is joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko.
Kevin Ford, Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros
(University of Illinois at Urbana-Champaign)
Kevin Ford, Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros
(University of Illinois at Urbana-Champaign)
Thursday, September 3, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 4, 2020 (1am AEST, 3am NZST)
Thursday, September 3, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, September 4, 2020 (1am AEST, 3am NZST)
Abstract: Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.
Abstract: Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.
Umberto Zannier, Torsion in elliptic familes and applications to billiards
(Scuola Normale Superiore Pisa)
Umberto Zannier, Torsion in elliptic familes and applications to billiards
(Scuola Normale Superiore Pisa)
Tuesday, September 1, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Tuesday, September 1, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)
Abstract: We shall consider elliptic pencils, of which the best-known example is probably the Legendre family L_t: y^2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on L_t depending on t) it is an issue to study the set of complex b such that P(b) is torsion on L_b. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,\pi), there are only finitely many cases for which both billiard trajectories are periodic.
Abstract: We shall consider elliptic pencils, of which the best-known example is probably the Legendre family L_t: y^2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on L_t depending on t) it is an issue to study the set of complex b such that P(b) is torsion on L_b. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,\pi), there are only finitely many cases for which both billiard trajectories are periodic.
Hector Pasten, A Chabauty-Coleman bound for hyperbolic surfaces in abelian threefolds
(Pontificia Universidad Católica de Chile)
Hector Pasten, A Chabauty-Coleman bound for hyperbolic surfaces in abelian threefolds
(Pontificia Universidad Católica de Chile)
Thursday, August 27, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 28, 2020 (1am AEST, 3am NZST)
Thursday, August 27, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 28, 2020 (1am AEST, 3am NZST)
Abstract: A celebrated result of Coleman gives a completely explicit version of Chabauty's finiteness theorem for rational points in hyperbolic curves over a number field, by a study of zeros of p-adic analytic functions. After several developments around this result, the problem of proving an analogous explicit bound for higher dimensional subvarieties of abelian varieties remains elusive. In this talk I'll sketch the proof of such a bound for hyperbolic surfaces contained in abelian threefolds. This is joint work with Jerson Caro.
Abstract: A celebrated result of Coleman gives a completely explicit version of Chabauty's finiteness theorem for rational points in hyperbolic curves over a number field, by a study of zeros of p-adic analytic functions. After several developments around this result, the problem of proving an analogous explicit bound for higher dimensional subvarieties of abelian varieties remains elusive. In this talk I'll sketch the proof of such a bound for hyperbolic surfaces contained in abelian threefolds. This is joint work with Jerson Caro.
Christopher Skinner, Solving diagonal diophantine equations over general p-adic fields
(Princeton University)
Christopher Skinner, Solving diagonal diophantine equations over general p-adic fields
(Princeton University)
Thursday, August 20, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 21, 2020 (1am AEST, 3am NZST)
Thursday, August 20, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 21, 2020 (1am AEST, 3am NZST)
Abstract: This talk will explain a proof that a system of r diagonal equations
Abstract: This talk will explain a proof that a system of r diagonal equations
a_{i,1} x_1^d + ...+ a_{i,s} x_s^d = 0 , i = 1,...,r
a_{i,1} x_1^d + ...+ a_{i,s} x_s^d = 0 , i = 1,...,r
with coefficients in a p-adic field K has a non-trivial solution in K if the number of variables s exceeds 3r^2d^2 (if p > 2) or 8r^2d^2 (if p=2). This is the first bound that holds uniformly for all p-adic fields K and that is polynomial in r or d. The methods -- and talk -- are elementary.
with coefficients in a p-adic field K has a non-trivial solution in K if the number of variables s exceeds 3r^2d^2 (if p > 2) or 8r^2d^2 (if p=2). This is the first bound that holds uniformly for all p-adic fields K and that is polynomial in r or d. The methods -- and talk -- are elementary.
Carl Pomerance, "Practical numbers"
(Dartmouth College)
Carl Pomerance, "Practical numbers"
(Dartmouth College)
Thursday, August 13, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 14, 2020 (1am AEST, 3am NZST)
Thursday, August 13, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 14, 2020 (1am AEST, 3am NZST)
Abstract: A practical number n is one where each number up to n can be expressed as a subset sum of n's positive divisors. It seems that Fibonacci was interested in them since they have the property that all fractions m/n with m < n can be written as a sum of distinct unit fractions with denominators dividing n. With similar considerations in mind, Srinivasan in 1948 coined the term "practical". There has been quite a lot of effort to study their distribution, effort which has gone hand in hand with the development of the anatomy of integers. After work of Tenenbaum, Saias, and Weingartner, we now know the "Practical Number Theorem": the number of practical numbers up to x is asymptotically cx/log x, where c= 1.33607.... In this talk I'll discuss some recent developments, including work of Thompson who considered the allied concept of phi-practical numbers n (the polynomial t^n-1 has divisors over the integers of every degree up to n) and the proof (joint with Weingartner) of a conjecture of Margenstern that each large odd number can be expressed as a sum of a prime and a practical number.
Abstract: A practical number n is one where each number up to n can be expressed as a subset sum of n's positive divisors. It seems that Fibonacci was interested in them since they have the property that all fractions m/n with m < n can be written as a sum of distinct unit fractions with denominators dividing n. With similar considerations in mind, Srinivasan in 1948 coined the term "practical". There has been quite a lot of effort to study their distribution, effort which has gone hand in hand with the development of the anatomy of integers. After work of Tenenbaum, Saias, and Weingartner, we now know the "Practical Number Theorem": the number of practical numbers up to x is asymptotically cx/log x, where c= 1.33607.... In this talk I'll discuss some recent developments, including work of Thompson who considered the allied concept of phi-practical numbers n (the polynomial t^n-1 has divisors over the integers of every degree up to n) and the proof (joint with Weingartner) of a conjecture of Margenstern that each large odd number can be expressed as a sum of a prime and a practical number.
Bjorn Poonen, "Tetrahedra with rational dihedral angles"
(MIT)
Bjorn Poonen, "Tetrahedra with rational dihedral angles"
(MIT)
Thursday, August 6, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 7, 2020 (1am AEST, 3am NZST)
Thursday, August 6, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 7, 2020 (1am AEST, 3am NZST)
Abstract: In 1895, Hill discovered a 1-parameter family of tetrahedra whose dihedral angles are all rational multiples of pi. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity. Many previous authors have solved polynomial equations in roots of unity, but never with more than 12 monomials, and the Conway-Jones polynomial has 105 monomials! I will explain the method we use to solve it and our discovery that the full classification consists of two 1-parameter families and an explicit finite list of sporadic tetrahedra.
Abstract: In 1895, Hill discovered a 1-parameter family of tetrahedra whose dihedral angles are all rational multiples of pi. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity. Many previous authors have solved polynomial equations in roots of unity, but never with more than 12 monomials, and the Conway-Jones polynomial has 105 monomials! I will explain the method we use to solve it and our discovery that the full classification consists of two 1-parameter families and an explicit finite list of sporadic tetrahedra.
Building on this work, we classify all configurations of vectors in R^3 such that the angle between each pair is a rational multiple of pi. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than 9 vectors is contained in a particular 15-vector configuration.
Building on this work, we classify all configurations of vectors in R^3 such that the angle between each pair is a rational multiple of pi. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than 9 vectors is contained in a particular 15-vector configuration.
This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein.
This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein.
Joseph H. Silverman, "More Tips on Keeping Secrets in a Post-Quantum World: Lattice-Based Cryptography"
(Brown University)
Joseph H. Silverman, "More Tips on Keeping Secrets in a Post-Quantum World: Lattice-Based Cryptography"
(Brown University)
Thursday, July 30, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 31, 2020 (1am AEST, 3am NZST)
Thursday, July 30, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 31, 2020 (1am AEST, 3am NZST)
Abstract: What do internet commerce, online banking, and updates to your phone apps have in common? All of them depend on modern public key cryptography for security. For example, there is the RSA cryptosystem that is used by many internet browsers, and there is the elliptic curve based ECDSA digital signature scheme that is used in many applications, including Bitcoin. All of these cryptographic construction are doomed if/when someone (NSA? Russia? China?) builds a full-scale operational quantum computer. It hasn't happened yet, as far as we know, but there are vast resources being thrown at the problem, and slow-but-steady progress is being made. So the search is on for cryptographic algorithms that are secure against quantum computers. The first part of my talk will be a mix of math and history and prognostication centered around the themes of quantum computers and public key cryptography. The second part will discuss cryptographic constructions based on hard lattice problems, which is one of the approaches being proposed to build a post-quantum cryptographic infrastructure.
Abstract: What do internet commerce, online banking, and updates to your phone apps have in common? All of them depend on modern public key cryptography for security. For example, there is the RSA cryptosystem that is used by many internet browsers, and there is the elliptic curve based ECDSA digital signature scheme that is used in many applications, including Bitcoin. All of these cryptographic construction are doomed if/when someone (NSA? Russia? China?) builds a full-scale operational quantum computer. It hasn't happened yet, as far as we know, but there are vast resources being thrown at the problem, and slow-but-steady progress is being made. So the search is on for cryptographic algorithms that are secure against quantum computers. The first part of my talk will be a mix of math and history and prognostication centered around the themes of quantum computers and public key cryptography. The second part will discuss cryptographic constructions based on hard lattice problems, which is one of the approaches being proposed to build a post-quantum cryptographic infrastructure.
Jordan Ellenberg, "What’s up in arithmetic statistics?"
(University of Wisconsin–Madison)
Jordan Ellenberg, "What’s up in arithmetic statistics?"
(University of Wisconsin–Madison)
Thursday, July 23, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 24, 2020 (1am AEST, 3am NZST)
Thursday, July 23, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 24, 2020 (1am AEST, 3am NZST)
Abstract: If not for a global pandemic, a bunch of mathematicians would have gathered in Germany to talk about what’s going on in the geometry of arithmetic statistics, which I would roughly describe as “methods from arithmetic geometry brought to bear on probabilistic questions about arithmetic objects". What does the maximal unramified extension of a random number field look like? What is the probability that a random elliptic curve has a 2-Selmer group of rank 100? How do you count points on a stack? I’ll give a survey of what’s happening in questions in this area, trying to emphasize open questions.
Abstract: If not for a global pandemic, a bunch of mathematicians would have gathered in Germany to talk about what’s going on in the geometry of arithmetic statistics, which I would roughly describe as “methods from arithmetic geometry brought to bear on probabilistic questions about arithmetic objects". What does the maximal unramified extension of a random number field look like? What is the probability that a random elliptic curve has a 2-Selmer group of rank 100? How do you count points on a stack? I’ll give a survey of what’s happening in questions in this area, trying to emphasize open questions.
Wadim Zudilin, "Irrationality through an irrational time"
(Radboud University Nijmegen)
Wadim Zudilin, "Irrationality through an irrational time"
(Radboud University Nijmegen)
Tuesday, July 21, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, July 21, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: After reviewing some recent development and achievements related to diophantine problems of the values of Riemann's zeta function and generalized polylogarithms (not all coming from myself!), I will move the focus to \pi=3.1415926... and its rational approximations. Specifically, I will discuss a construction of rational approximations to the number that leads to the record irrationality measure of \pi. The talk is based on joint work with Doron Zeilberger.
Abstract: After reviewing some recent development and achievements related to diophantine problems of the values of Riemann's zeta function and generalized polylogarithms (not all coming from myself!), I will move the focus to \pi=3.1415926... and its rational approximations. Specifically, I will discuss a construction of rational approximations to the number that leads to the record irrationality measure of \pi. The talk is based on joint work with Doron Zeilberger.
Jennifer Balakrishnan, "A tale of three curves"
(Boston University)
Jennifer Balakrishnan, "A tale of three curves"
(Boston University)
Thursday, July 16, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 17, 2020 (1am AEST, 3am NZST)
Thursday, July 16, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 17, 2020 (1am AEST, 3am NZST)
Abstract: We will describe variants of the Chabauty-Coleman method and quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
Abstract: We will describe variants of the Chabauty-Coleman method and quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
Ken Ono, "Variants of Lehmer's speculation for newforms"
(University of Virginia)
Ken Ono, "Variants of Lehmer's speculation for newforms"
(University of Virginia)
Monday, July 13, 2020 (5pm PDT, 8pm EDT)
Tuesday, July 14, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Monday, July 13, 2020 (5pm PDT, 8pm EDT)
Tuesday, July 14, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Abstract: In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of τ(n), or is a Fourier coefficient of any given newform. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained some results that will be described here. For example, infinitely many spaces are presented for which the primes ℓ≤37 are not absolute values of coefficients of any newforms with integer coefficients. For Ramanujan’s tau-function, such results imply, for n>1, that
Abstract: In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of τ(n), or is a Fourier coefficient of any given newform. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained some results that will be described here. For example, infinitely many spaces are presented for which the primes ℓ≤37 are not absolute values of coefficients of any newforms with integer coefficients. For Ramanujan’s tau-function, such results imply, for n>1, that
τ(n)∉{±1,±3,±5,±7,±13,±17,−19,±23,±37,±691}.
τ(n)∉{±1,±3,±5,±7,±13,±17,−19,±23,±37,±691}.
Lillian Pierce, "On Bourgain’s counterexample for the Schrödinger maximal function"
(Duke University)
Lillian Pierce, "On Bourgain’s counterexample for the Schrödinger maximal function"
(Duke University)
Thursday, July 9, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 10, 2020 (1am AEST, 3am NZST)
Thursday, July 9, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 10, 2020 (1am AEST, 3am NZST)
Abstract: There is a long and visible history of applications of analytic methods to number theory. More recently we are starting to recognize applications of number-theoretic methods to analysis. In this talk we will describe an important recent application in this direction.
Abstract: There is a long and visible history of applications of analytic methods to number theory. More recently we are starting to recognize applications of number-theoretic methods to analysis. In this talk we will describe an important recent application in this direction.
In 1980, Carleson asked a question in PDE's: for what class of initial data functions does a pointwise a.e. convergence result hold for the solution of the linear Schrödinger equation? Over the next decades, many people developed counterexamples to show “necessary conditions,” and on the other hand positive results to show “sufficient conditions.” In 2016 Bourgain wrote a 3-page paper using facts from number theory to construct a family of counterexamples. A 2019 Annals paper of Du and Zhang then resolved the question by proving positive results that push the “sufficient conditions” to meet Bourgain’s “necessary conditions."
In 1980, Carleson asked a question in PDE's: for what class of initial data functions does a pointwise a.e. convergence result hold for the solution of the linear Schrödinger equation? Over the next decades, many people developed counterexamples to show “necessary conditions,” and on the other hand positive results to show “sufficient conditions.” In 2016 Bourgain wrote a 3-page paper using facts from number theory to construct a family of counterexamples. A 2019 Annals paper of Du and Zhang then resolved the question by proving positive results that push the “sufficient conditions” to meet Bourgain’s “necessary conditions."
Bourgain’s construction was regarded as somewhat mysterious. In this talk, we give an overview of how to rigorously derive Bourgain’s construction using ideas from number theory. Our strategy is to start from “zero knowledge" and gradually optimize the set-up to arrive at the final counterexample. This talk will be broadly accessible.
Bourgain’s construction was regarded as somewhat mysterious. In this talk, we give an overview of how to rigorously derive Bourgain’s construction using ideas from number theory. Our strategy is to start from “zero knowledge" and gradually optimize the set-up to arrive at the final counterexample. This talk will be broadly accessible.
René Schoof, "Abelian varieties over Q(\sqrt{97}) with good reduction everywhere"
(Università di Roma “Tor Vergata”)
René Schoof, "Abelian varieties over Q(\sqrt{97}) with good reduction everywhere"
(Università di Roma “Tor Vergata”)
Tuesday, July 7, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, July 7, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: Under assumption of the Generalized Riemann Hypothesis we show that every abelian variety over Q(\sqrt{97}) with good reduction everywhere is isogenous to a power of a certain 3-dimensional modular abelian variety.
Abstract: Under assumption of the Generalized Riemann Hypothesis we show that every abelian variety over Q(\sqrt{97}) with good reduction everywhere is isogenous to a power of a certain 3-dimensional modular abelian variety.
(joint with Lassina Dembele)
(joint with Lassina Dembele)
James Maynard, "Primes in arithmetic progressions to large moduli"
(University of Oxford)
James Maynard, "Primes in arithmetic progressions to large moduli"
(University of Oxford)
Thursday, July 2, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 3, 2020 (1am AEST, 3am NZST)
Thursday, July 2, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 3, 2020 (1am AEST, 3am NZST)
Abstract: How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis!
Abstract: How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis!
If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.
If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.
Kannan Soundararajan, "Equidistribution from the Chinese Remainder Theorem"
(Stanford University)
Kannan Soundararajan, "Equidistribution from the Chinese Remainder Theorem"
(Stanford University)
Monday, June 29, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 30, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Monday, June 29, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 30, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Abstract: Suppose for each prime p we are given a set A_p (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set A_q of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in A_q will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors. (Joint work with Emmanuel Kowalski.)
Abstract: Suppose for each prime p we are given a set A_p (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set A_q of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in A_q will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors. (Joint work with Emmanuel Kowalski.)
Peter Sarnak, "Integer points on affine cubic surfaces"
(Institute for Advanced Study and Princeton University)
Peter Sarnak, "Integer points on affine cubic surfaces"
(Institute for Advanced Study and Princeton University)
Thursday, June 25, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 26, 2020 (1am AEST, 3am NZST)
Thursday, June 25, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 26, 2020 (1am AEST, 3am NZST)
Abstract: The level sets of a cubic polynomial in four or more variables tends to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in say a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in SL_2 over such rings.
Abstract: The level sets of a cubic polynomial in four or more variables tends to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in say a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in SL_2 over such rings.
Igor Shparlinski, "Weyl Sums: Large, Small and Typical"
(UNSW Sydney)
Igor Shparlinski, "Weyl Sums: Large, Small and Typical"
(UNSW Sydney)
Tuesday, June 23, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, June 23, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: While Vinogradov’s Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments of the Weyl sums
Abstract: While Vinogradov’s Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments of the Weyl sums
S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+…+u_dn^d))
S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+…+u_dn^d))
where u = (u_1,...,u_d) \in [0,1)^d, very little is known about the distribution, or even existence, of u \in [0,1)^d, for which these sums are very large, or small, or close to their average value N^{1/2}. In this talk, we describe recent progress towards these and some related questions.
where u = (u_1,...,u_d) \in [0,1)^d, very little is known about the distribution, or even existence, of u \in [0,1)^d, for which these sums are very large, or small, or close to their average value N^{1/2}. In this talk, we describe recent progress towards these and some related questions.
We also present some new bounds on S(u;N) which hold for almost all (u_i)_{i\in I} and all (u_j)_{j\in J}, where I \cup J is a partition of {1,…,,d}. These bounds improve similar results of T. Wooley (2015). Our method also applies to binomial sums
We also present some new bounds on S(u;N) which hold for almost all (u_i)_{i\in I} and all (u_j)_{j\in J}, where I \cup J is a partition of {1,…,,d}. These bounds improve similar results of T. Wooley (2015). Our method also applies to binomial sums
T(x,y; N) = \sum_{1\le n \le N} \exp(2 \pi i (xn+yn^d))
T(x,y; N) = \sum_{1\le n \le N} \exp(2 \pi i (xn+yn^d))
with x,y \in [0,1), in which case we improve some results of M.B. Erdogan and G. Shakan (2019).
with x,y \in [0,1), in which case we improve some results of M.B. Erdogan and G. Shakan (2019).
This is a joint work with Changhao Chen and Bryce Kerr.
This is a joint work with Changhao Chen and Bryce Kerr.
Elon Lindenstrauss, "Diagonalizable flows, joinings, and arithmetic applications"
(Hebrew University of Jerusalem)
Elon Lindenstrauss, "Diagonalizable flows, joinings, and arithmetic applications"
(Hebrew University of Jerusalem)
Thursday, June 18, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 19, 2020 (1am AEST, 3am NZST)
Thursday, June 18, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 19, 2020 (1am AEST, 3am NZST)
Abstract: Rigidity properties of higher rank diagonalizable actions have proved to be powerful tools in understanding the distribution properties of rational tori in arithmetic quotients. Perhaps the simplest, and best known, example of such an equidistribution question is the equidistribution of CM points of a given discriminant on the modular curve. The equidistribution of CM points was established by Duke using analytic methods, but for finer questions (and questions regarding equidistribution on higher rank spaces) the ergodic theoretic approach has proved to be quite powerful.
Abstract: Rigidity properties of higher rank diagonalizable actions have proved to be powerful tools in understanding the distribution properties of rational tori in arithmetic quotients. Perhaps the simplest, and best known, example of such an equidistribution question is the equidistribution of CM points of a given discriminant on the modular curve. The equidistribution of CM points was established by Duke using analytic methods, but for finer questions (and questions regarding equidistribution on higher rank spaces) the ergodic theoretic approach has proved to be quite powerful.
I will survey some of the results in this direction, including several results about joint distributions of collections of points in product spaces by Aka, Einsiedler, Khayutin, Shapira, Wieser and other researchers.
I will survey some of the results in this direction, including several results about joint distributions of collections of points in product spaces by Aka, Einsiedler, Khayutin, Shapira, Wieser and other researchers.
Harald Andrés Helfgott, "Optimality of the logarithmic upper-bound sieve, with explicit estimates"
(Göttingen/CNRS (IMJ))
Harald Andrés Helfgott, "Optimality of the logarithmic upper-bound sieve, with explicit estimates"
(Göttingen/CNRS (IMJ))
Tuesday, June 16, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, June 16, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: At the simplest level, an upper bound sieve of Selberg type is a choice of rho(d), d<=D, with rho(1)=1, such that
Abstract: At the simplest level, an upper bound sieve of Selberg type is a choice of rho(d), d<=D, with rho(1)=1, such that
S = \sum_{n\leq N} \left(\sum_{d|n} \mu(d) \rho(d)\right)^2
S = \sum_{n\leq N} \left(\sum_{d|n} \mu(d) \rho(d)\right)^2
is as small as possible.
is as small as possible.
The optimal choice of rho(d) for given D was found by Selberg. However, for several applications, it is better to work with functions rho(d) that are scalings of a given continuous or monotonic function eta. The question is then what is the best function eta, and how does S for given eta and D compares to S for Selberg's choice.
The optimal choice of rho(d) for given D was found by Selberg. However, for several applications, it is better to work with functions rho(d) that are scalings of a given continuous or monotonic function eta. The question is then what is the best function eta, and how does S for given eta and D compares to S for Selberg's choice.
The most common choice of eta is that of Barban-Vehov (1968), which gives an S with the same main term as Selberg's S. We show that Barban and Vehov's choice is optimal among all eta, not just (as we knew) when it comes to the main term, but even when it comes to the second-order term, which is negative and which we determine explicitly.
The most common choice of eta is that of Barban-Vehov (1968), which gives an S with the same main term as Selberg's S. We show that Barban and Vehov's choice is optimal among all eta, not just (as we knew) when it comes to the main term, but even when it comes to the second-order term, which is negative and which we determine explicitly.
This is joint work with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero.
This is joint work with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero.
Yuri Bilu, "Trinomials, singular moduli and Riffaut's conjecture"
(University of Bordeaux)
Yuri Bilu, "Trinomials, singular moduli and Riffaut's conjecture"
(University of Bordeaux)
Thursday, June 11, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 12, 2020 (1am AEST, 3am NZST)
Thursday, June 11, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 12, 2020 (1am AEST, 3am NZST)
Abstract: Riffaut (2019) conjectured that a singular modulus of degree h>2 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results. A joint work with Florian Luca and Amalia Pizarro.
Abstract: Riffaut (2019) conjectured that a singular modulus of degree h>2 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results. A joint work with Florian Luca and Amalia Pizarro.
Felipe Voloch, "Value sets of sparse polynomials"
(University of Canterbury)
Felipe Voloch, "Value sets of sparse polynomials"
(University of Canterbury)
Monday, June 8, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 9, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Monday, June 8, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 9, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Abstract: We obtain a lower bound on the size of the value set f(F_p) of a sparse polynomial f(x) in F_p[x] over a finite field of p elements when p is prime. This bound is uniform with respect to the degree and depends on the number of terms of f.
Abstract: We obtain a lower bound on the size of the value set f(F_p) of a sparse polynomial f(x) in F_p[x] over a finite field of p elements when p is prime. This bound is uniform with respect to the degree and depends on the number of terms of f.
Timothy Browning, "Random Diophantine equations"
(IST Austria)
Timothy Browning, "Random Diophantine equations"
(IST Austria)
Thursday, June 4, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 5, 2020 (1am AEST, 3am NZST)
Thursday, June 4, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 5, 2020 (1am AEST, 3am NZST)
Abstract: I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers.
Abstract: I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers.
While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.
While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.
Kaisa Matomäki, "Multiplicative functions in short intervals revisited"
(University of Turku)
Kaisa Matomäki, "Multiplicative functions in short intervals revisited"
(University of Turku)
Tuesday, June 2, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, June 2, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: A few years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals [x, x+h] is close to its average on a long interval [x, 2x]. This result has since been utilized in many applications.
Abstract: A few years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals [x, x+h] is close to its average on a long interval [x, 2x]. This result has since been utilized in many applications.
In a work in progress that I will talk about, Radziwill and I revisit the problem and generalise our result to functions which vanish often as well as prove a power-saving upper bound for the number of exceptional intervals (i.e. we show that there are O(X/h^\kappa) exceptional x \in [X, 2X]).
In a work in progress that I will talk about, Radziwill and I revisit the problem and generalise our result to functions which vanish often as well as prove a power-saving upper bound for the number of exceptional intervals (i.e. we show that there are O(X/h^\kappa) exceptional x \in [X, 2X]).
We apply this result for instance to studying gaps between norm forms of an arbitrary number field.
We apply this result for instance to studying gaps between norm forms of an arbitrary number field.
Trevor Wooley, "Bracket quadratics, Hua’s Lemma and Vinogradov’s mean value theorem"
(Purdue University)
Trevor Wooley, "Bracket quadratics, Hua’s Lemma and Vinogradov’s mean value theorem"
(Purdue University)
Thursday, May 28, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 29, 2020 (1am AEST, 3am NZST)
Thursday, May 28, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 29, 2020 (1am AEST, 3am NZST)
Abstract: A little over a decade ago, Ben Green posed the problem of showing that all large integers are the sum of at most a bounded number of bracket quadratic polynomials of the shape n[n\theta], for natural numbers n, in which \theta is an irrational number such as the square-root of 2. This was resolved in the PhD thesis of Vicky Neale, although no explicit bound was given concerning the number of variables required to achieve success. In this talk we describe a version of Hua’s lemma for this problem that can be applied via the Hardy-Littlewood method to obtain a conclusion with 5 variables. The associated argument differs according to whether \theta is a quadratic irrational or not. We also explain how related versions of Hua’s lemma may be interpreted in terms of discrete restriction variants of Vinogradov’s mean value theorem, thus providing a route to generalisation.
Abstract: A little over a decade ago, Ben Green posed the problem of showing that all large integers are the sum of at most a bounded number of bracket quadratic polynomials of the shape n[n\theta], for natural numbers n, in which \theta is an irrational number such as the square-root of 2. This was resolved in the PhD thesis of Vicky Neale, although no explicit bound was given concerning the number of variables required to achieve success. In this talk we describe a version of Hua’s lemma for this problem that can be applied via the Hardy-Littlewood method to obtain a conclusion with 5 variables. The associated argument differs according to whether \theta is a quadratic irrational or not. We also explain how related versions of Hua’s lemma may be interpreted in terms of discrete restriction variants of Vinogradov’s mean value theorem, thus providing a route to generalisation.
Bas Edixhoven, "Geometric quadratic Chabauty"
(Leiden University)
Bas Edixhoven, "Geometric quadratic Chabauty"
(Leiden University)
Tuesday, May 26, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, May 26, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: Joint work with Guido Lido (see arxiv preprint). Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). Our work gives a version of this method that uses only `simple algebraic geometry' (line bundles over the jacobian and models over the integers). For the talk, no knowledge of all this algebraic geometry is required, it will be accessible to all number theorists.
Abstract: Joint work with Guido Lido (see arxiv preprint). Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). Our work gives a version of this method that uses only `simple algebraic geometry' (line bundles over the jacobian and models over the integers). For the talk, no knowledge of all this algebraic geometry is required, it will be accessible to all number theorists.
Zeev Rudnick, "Prime lattice points in ovals"
(Tel-Aviv University)
Zeev Rudnick, "Prime lattice points in ovals"
(Tel-Aviv University)
Thursday, May 21, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 22, 2020 (1am AEST, 3am NZST)
Thursday, May 21, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 22, 2020 (1am AEST, 3am NZST)
Abstract: The study of the number of lattice points in dilated regions has a long history, with several outstanding open problems. In this lecture, I will describe a new variant of the problem, in which we study the distribution of lattice points with prime coordinates. We count lattice points in which both coordinates are prime, suitably weighted, which lie in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. We obtain an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term. (joint work with Bingrong Huang).
Abstract: The study of the number of lattice points in dilated regions has a long history, with several outstanding open problems. In this lecture, I will describe a new variant of the problem, in which we study the distribution of lattice points with prime coordinates. We count lattice points in which both coordinates are prime, suitably weighted, which lie in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. We obtain an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term. (joint work with Bingrong Huang).
Kristin Lauter, "How to Keep your Secrets in a Post-Quantum World"
(Microsoft Research Redmond Labs)
Kristin Lauter, "How to Keep your Secrets in a Post-Quantum World"
(Microsoft Research Redmond Labs)
Monday, May 18, 2020 (5pm PDT, 8pm EDT)
Tuesday, May 19, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Monday, May 18, 2020 (5pm PDT, 8pm EDT)
Tuesday, May 19, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)
Abstract: As we move towards a world which includes quantum computers which exist at scale, we are forced to consider the question of what hard problems in mathematics our next generation of cryptographic systems will be based on. Supersingular Isogeny Graphs were proposed for use in cryptography in 2006 by Charles, Goren, and Lauter. Supersingular Isogeny Graphs are examples of Ramanujan graphs, which are optimal expander graphs. These graphs have the property that relatively short walks on the graph approximate the uniform distribution, and for this reason, walks on expander graphs are often used as a good source of randomness in computer science. But the reason these graphs are important for cryptography is that finding paths in these graphs, i.e. routing, is hard: there are no known subexponential algorithms to solve this problem, either classically or on a quantum computer. For this reason, cryptosystems based on the hardness of problems on Supersingular Isogeny Graphs are currently under consideration for standardization in the NIST Post-Quantum Cryptography (PQC) Competition. This talk will introduce these graphs, the cryptographic applications, and the various algorithmic approaches which have been tried to attack these systems.
Abstract: As we move towards a world which includes quantum computers which exist at scale, we are forced to consider the question of what hard problems in mathematics our next generation of cryptographic systems will be based on. Supersingular Isogeny Graphs were proposed for use in cryptography in 2006 by Charles, Goren, and Lauter. Supersingular Isogeny Graphs are examples of Ramanujan graphs, which are optimal expander graphs. These graphs have the property that relatively short walks on the graph approximate the uniform distribution, and for this reason, walks on expander graphs are often used as a good source of randomness in computer science. But the reason these graphs are important for cryptography is that finding paths in these graphs, i.e. routing, is hard: there are no known subexponential algorithms to solve this problem, either classically or on a quantum computer. For this reason, cryptosystems based on the hardness of problems on Supersingular Isogeny Graphs are currently under consideration for standardization in the NIST Post-Quantum Cryptography (PQC) Competition. This talk will introduce these graphs, the cryptographic applications, and the various algorithmic approaches which have been tried to attack these systems.
Valentin Blomer, "Joint equidistribution and fractional moments of L-functions"
(Universität Bonn)
Valentin Blomer, "Joint equidistribution and fractional moments of L-functions"
(Universität Bonn)
Thursday, May 14, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Thursday, May 14, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Abstract: Integral points on spheres of large radius D^(1/2) equidstribute (subject to appropriate congruence conditions), and so do Heegner points of large discriminant D on the modular curve. Both sets have roughly the same cardinality, and there is a natural way to associate with each point on the sphere a Heegner point. Do these pairs equidistribute in the product space of the sphere and the modular curve as D tends to infinity?
Abstract: Integral points on spheres of large radius D^(1/2) equidstribute (subject to appropriate congruence conditions), and so do Heegner points of large discriminant D on the modular curve. Both sets have roughly the same cardinality, and there is a natural way to associate with each point on the sphere a Heegner point. Do these pairs equidistribute in the product space of the sphere and the modular curve as D tends to infinity?
A seemingly very different, but structurally similar joint equidistribution problem can be asked for the supersingular reduction at two different primes of elliptic curves with CM by an order of large discriminant D.
A seemingly very different, but structurally similar joint equidistribution problem can be asked for the supersingular reduction at two different primes of elliptic curves with CM by an order of large discriminant D.
Both equidistribution problems have been studied by ergodic methods under certain conditions on D. I will explain how to use number theory and families of high degree L-functions to obtain an effective equidistribution statement with a rate of convergence, assuming GRH. This is joint work in progress with F. Brumley.
Both equidistribution problems have been studied by ergodic methods under certain conditions on D. I will explain how to use number theory and families of high degree L-functions to obtain an effective equidistribution statement with a rate of convergence, assuming GRH. This is joint work in progress with F. Brumley.
Michel Waldschmidt, "Representation of integers by cyclotomic binary forms"
(Sorbonne University)
Michel Waldschmidt, "Representation of integers by cyclotomic binary forms"
(Sorbonne University)
Tuesday, May 12, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Tuesday, May 12, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)
Abstract: The representation of positive integers as a sum of two squares is a classical problem studied by Landau and Ramanujan. A similar result has been obtained by Bernays for positive definite binary form. In joint works with Claude Levesque and Etienne Fouvry, we consider the representation of integers by the binary forms which are deduced from the cyclotomic polynomials. One main tool is a recent result of Stewart and Xiao which generalizes the theorem of Bernays to binary forms of higher degree.
Abstract: The representation of positive integers as a sum of two squares is a classical problem studied by Landau and Ramanujan. A similar result has been obtained by Bernays for positive definite binary form. In joint works with Claude Levesque and Etienne Fouvry, we consider the representation of integers by the binary forms which are deduced from the cyclotomic polynomials. One main tool is a recent result of Stewart and Xiao which generalizes the theorem of Bernays to binary forms of higher degree.
Andrew Sutherland, "Sums of three cubes"
(MIT)
Andrew Sutherland, "Sums of three cubes"
(MIT)
Thursday, May 7, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Thursday, May 7, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Abstract: In 1953 Mordell asked whether one can represent 3 as a sum of three cubes in any way other than 1^3+1^3+1^3 and 4^3+4^3 -5^3. Mordell's question spurred many computational investigations over the years, and while none found a new solution for 3, they eventually determined which of the first 100 positive integers k can be represented as a sum of three cubes in all but one case: k=42.
Abstract: In 1953 Mordell asked whether one can represent 3 as a sum of three cubes in any way other than 1^3+1^3+1^3 and 4^3+4^3 -5^3. Mordell's question spurred many computational investigations over the years, and while none found a new solution for 3, they eventually determined which of the first 100 positive integers k can be represented as a sum of three cubes in all but one case: k=42.
In this talk I will present joint work with Andrew Booker that used Charity Engine's crowd-sourced compute grid to affirmatively answer Mordell's question, as well as settling the case k=42. I will also discuss a conjecture of Heath-Brown that predicts the existence of infinitely many more solutions and also explains why they are so difficult to find
In this talk I will present joint work with Andrew Booker that used Charity Engine's crowd-sourced compute grid to affirmatively answer Mordell's question, as well as settling the case k=42. I will also discuss a conjecture of Heath-Brown that predicts the existence of infinitely many more solutions and also explains why they are so difficult to find
Andrew Granville, "Frobenius's postage stamp problem, and beyond..."
(Université de Montréal)
Andrew Granville, "Frobenius's postage stamp problem, and beyond..."
(Université de Montréal)
Thursday, April 30, 2020 (8am PDT, 11am EDT, 5pm CEST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Thursday, April 30, 2020 (8am PDT, 11am EDT, 5pm CEST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)
Abstract: We study this famous old problem from the modern perspective of additive combinatorics, and then look at generalizations.
Abstract: We study this famous old problem from the modern perspective of additive combinatorics, and then look at generalizations.