Previous Talks

Andreas Strömbergsson, An effective equidistribution result in the space of 2-dimensional tori with k marked points
(Uppsala University) 

Thursday, March 27, 2025 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

Friday, March 28, 2025 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Let X be the homogeneous space Gamma\G, where G is the semidirect product of SL(2,R) and a direct sum of k copies of R^2, and where Gamma is the subgroup of integer elements in G. I will present a result giving effective equidistribution of one-parameter unipotent orbits in the space X. The non-effective version of this result is a special case of Ratner's celebrated equidistribution theorem for unipotent flows in homogeneous dynamics, and the particular setting which we consider has several known applications to problems in number theory and mathematical physics. Our proof makes use of the delta method in the form developed by Heath-Brown. This is joint work with Anders Södergren and Pankaj Vishe.

Link to recording (YouTube)

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Gergely Harcos, A new zero-free region for Rankin–Selberg L-functions
(Alfréd Rényi Institute of Mathematics) 

Thursday, March 20, 2025 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 21, 2025 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will present a new zero-free region for all GL(1)-twists of GL(m)×GL(n) Rankin–Selberg L-functions. The proof is inspired by Siegel’s celebrated lower bound for Dirichlet L-functions at s=1. I will also discuss some applications. Joint work with Jesse Thorner.

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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)

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) 

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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) 

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.

Link to recording (YouTube)

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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)

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.

Link to recording (YouTube)

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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)

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.

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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)

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.

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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)

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)

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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)

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)

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. 

Link to recording (YouTube)

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)

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. 

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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)

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.

Link to recording (YouTube)

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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)

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)

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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)

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.

Link to recording (YouTube)

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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)

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.

Link to recording (YouTube)

Link to slides

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)

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)

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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)

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)

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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)

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)

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.

Joint w. Ben Green (Oxford)

Link to recording (YouTube)

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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)

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)

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).  

Link to recording (YouTube)

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)

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.

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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)

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

$$\sum_{\chi(q)} L(\chi^{a_1},1/2)\cdots L(\chi^a_k,1/2)$$

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 $\pm 1$).

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.

This is joint work with E. Fouvry, E. Kowalski and W. Sawin.

Link to recording (YouTube)

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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)

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

$$\zeta_{\theta}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{N_k(\theta)}{k}z^k\right).$$

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.

Link to recording (YouTube)

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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)

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.

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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)

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.

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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)

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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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.

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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)

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. 

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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)

Abstract: Link

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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)

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. 

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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)

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.

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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)

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.

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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)

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.

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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)

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.)

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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)

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.

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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)

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.

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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)

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.

No background in quantum computation will be assumed.

Based on the arXiv preprint: https://arxiv.org/abs/2308.06572

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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)

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.

Link to slides

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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)

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.

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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)

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. 

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.

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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)

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. 

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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)

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.

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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)

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)

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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)

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.

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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)

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

          { 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.)

Link to slides

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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)

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)

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.

Link to recording (YouTube)

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)

Link to abstract

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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)

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.

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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)

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.

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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)

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.

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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)

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.

This is based on joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.

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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)

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 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:

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.

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.

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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)

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. 

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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)

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.

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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)

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. 

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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)

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. 

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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)

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. 

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. 

Based on a joint work with Alexei Entin and Eilidh McKemmie

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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)

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. 

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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)

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. 

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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)

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. 

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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)

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.

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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)

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. 

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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)

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.