Previous Talks

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)

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.


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)

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.

Link to slides

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)

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.

Link to slides

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)

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 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.

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)


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.

This is joint work with Frank Calegari and Vesselin Dimitrov.

Link to slides

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)

Abstract: Link

Link to slides

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)

Abstract: Link

Link to slides

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)

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:

1. At least a sixth of integers are not the sum of two rational cubes,

and

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.)

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)


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.

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

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)

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.

Link to slides

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)

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.

Link to slides

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)

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?

Link to slides

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)

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)


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.

Link to slides

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)

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.

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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).

Link to slides

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)

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.

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.

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.

Link to slides

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)

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.

Link to slides

Link to website

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)

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:

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 $n$?

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.

Link to slides

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)

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.

Abstract: Link to PDF

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)

Abstract: Link to PDF

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)


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.

Link to slides

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)

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)


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).


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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides

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)

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.


Link to slides

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)

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?

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.

Link to slides

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)

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)


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.

Link to slides

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)

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)


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.

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.

This is joint work with Sebastián Herrero (PUC Valparaíso) and Juan Rivera-Letelier (Rochester).

Link to slides

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)

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.

Link to slides

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)


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 $n$.

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)

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.

Link to slides

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)

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)


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.

Link to slides

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)

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.

Link to slides

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)

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.

Link to slides


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)

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.

Link to slides

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)

Abstract: We discuss recent progress on understanding connections between the objects in the title.

Link to slides

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)

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)

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)

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)


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.

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.

Link to slides

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)

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.

Link to slides

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)

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).

Link to slides

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)

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.

Link to slides

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)

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,×).

The talk is based on recent joint work with Florian Richter.

Link to slides

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)

Friday, March 19, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)

Abstract: Link

Link to slides

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)

Friday, March 12, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)

Abstract: Link

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)

Friday, March 5, 2021 (1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT)

Abstract: Link

Link to slides

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)

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.

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.

Link to slides

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)

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.


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)

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.

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)

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.

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)

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.

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)


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)

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. 

Q1: What is the probability that A is irreducible as the degree goes to infinity?

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. 

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


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)

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)


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.

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)

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)

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.

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.

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)

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)

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.

Jacob Tsimerman, Bounding torsion in class group and families of local systems (University of Toronto)

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)

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)


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.

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.

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)

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.

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