Previous Talks

Matilde Lalín, Distributions of sums of the divisor function over function fields
(Université de Montréal)

Thursday, February 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 3, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick studied the mean-square of sums of the divisor function $d_k(f)$  over short intervals and over arithmetic progressions for the function field  $\mathbb{F}_q[T]$. By results from the Katz and Sarnak philosophy, they were able to relate these problems to certain integrals over the ensemble of unitary matrices when  $q$ goes to infinity. We study similar problems leading to integrals over the ensembles of symplectic and orthogonal matrices when $q$ goes to infinity. This is joint work with Vivian Kuperberg.

Link to recording

Link to slides

Robert Wilms, On equidistribution in Arakelov theory
(University of Basel)

Thursday, January 26, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 27, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: As a motivating example of its own interest I will first discuss a new equidistribution result for the zero sets of integer polynomials. More precisely, I will give a condition such that the zero sets tends to equidistribute with respect to the Fubini-Study measure and I will show that this condition is generically satisfied in sets of polynomials of bounded Bombieri norm. In the second part, I will embed this example in a much more general framework about the distribution of the divisors of small sections of arithmetically ample hermitian line bundles in Arakelov theory.

Link to recording

Link to slides

Cécile Dartyge, On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases
(Institut  Élie Cartan, Université de Lorraine)

Thursday, January 19, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 20, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Let P(X) be a monic, quartic, irreducible polynomial of Z[X] with cyclic or dihedral Galois group. We prove that there exists c_P >0, such that for a positive proportion of integers n, P(n) has a prime factor bigger than n^{1+c_P}.

This is a joint work with James Maynard.

Link to recording

Link to slides

Régis de la Bretèche, Higher moments of primes  in arithmetic progressions
(Institut de Mathématiques de Jussieu-Paris Rive Gauche)

Thursday, January 12, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 13, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In a joint work with Daniel Fiorilli, we develop a new method to prove lower bounds of some moments related to the distribution of primes in arithmetic progressions. We shall present  main results and explain some aspects of the proofs. To prove our results, we assume GRH but we succeed to avoid linearly independence on zeroes hypothesis.

Link to recording

Link to slides

Umberto Zannier, Bounded generation in linear groups and exponential parametrizations
(Scuola Normale Superiore Pisa)

Thursday, December 22, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 23, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of “bounded generation” in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on “anisotropic groups” (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields)  are “sparse”. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.

William Duke, On the analytic theory of isotropic ternary quadratic forms
(UCLA)

Thursday, December 15, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 16, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is explicitly computed in terms of local data determined by the orbit.  This is compared  with the well-known asymptotic for the  count of all primitive zeros. Together with an extension  of results of Kneser by R. Schulze-Pillot on the classes  in a genus of representations, this yields a formula for the number of  orbits,  summed over a genus of forms,  in terms of the number of local orbits. For a certain special class of forms a simple explicit formula is given for this number.

Link to recording

Link to slides

Laura DeMarco, Lattès maps, bifurcations, and arithmetic
(Harvard University)

Thursday, December 8, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 9, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In the field of holomorphic dynamics, we learn that the Lattès maps -- the rational functions on P^1 that are quotients of maps on elliptic curves -- are rather boring.  We can understand their dynamics completely.  But viewed arithmetically, there are still unanswered questions.  I'll begin the talk with some history of these maps.  Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on P^1 and, in turn, new perspectives on the arithmetic side.  The new material is a joint project with Myrto Mavraki.

Robert J. Lemke Oliver, Uniform exponent bounds on the number of primitive extensions of number fields
(Tufts University)

Thursday, December 1, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 2, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$.  This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach.

Here, we make progress on this weaker conjecture (which we term the ``uniform exponent conjecture'') in two ways.  First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation.  Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions.  These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank.  This is forthcoming work that grew out of conversations with M. Bhargava.

Link to slides

Jared Duker Lichtman, A proof of the Erdős primitive set conjecture
(University of Oxford)

Thursday, November 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 25, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In 1988 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.

Link to slides

Trevor Wooley, Waring’s Problem
(Purdue University)

Thursday, November 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 18, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 1770, E. Waring made an assertion these days interpreted as conjecturing that when $k$ is a natural number, all positive integers may be written as the sum of a number $g(k)$ of positive integral $k$-th powers, with $g(k)$ finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding $G(k)$, the least number $s$ having the property that all sufficiently large integers can be written as the sum of $s$ positive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange), $G(3)\le 7$ (Linnik), $G(4)=16$ (Davenport), and $G(5)\le 17$, $G(6)\le 24$, ..., $G(20)\le 142$ (Vaughan and Wooley). For large $k$ one has $G(k)\le k(\log k+\log \log k+2+o(1))$ (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.

Link to slides

Emanuel Carneiro, Hilbert spaces and low-lying zeros of L-functions
(ICTP)


Thursday, November 10, 2022 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 11, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In this talk I would like to present some ideas behind a general Hilbert space framework for solving certain optimization problems that arise when studying the distribution of the low-lying zeros of families of $L$-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.

Link to slides

Shai Evra, Optimal strong approximation and the Sarnak-Xue density hypothesis
(Hebrew University of Jerusalem)


Thursday, November 3, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 4, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: It is a classical result that the modulo map from SL_2(Z) to SL_2(Z/qZ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL_2(Z/qZ) has a lift in SL_2(Z) with coefficients of size at most q^e? A simple pigeonhole principle shows that e > 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL_2(Z). The proof relies on a density theorem of the Ramanujan conjecture for SL_2(Z).

In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands program.

This talk is based on ongoing joint works with B. Feigon, M. Gerbelli-Gauthier, H. Gustafssun, K. Maurischat and O. Parzanchevski.

Link to slides

Evelina Viada, Rational points on curves in a product of elliptic curves
(University of Göttingen)


Thursday, October 27, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 28, 2022 (2am AEDT, 4am NZST)

Abstract: The Mordell-Conjecture (Faltings Theorem) states that an algebraic curve of genus at least 2 has only finitely many rational points. The Torsion Anomalous Conjecture (TAC) generalises Faltings Theorem. In some cases the proofs of the TAC are effective, implying effective cases of the Mordell-Conjecture. I would like to explain an effective method to determine the K-rational points on certain families of curves and to present some new specific examples. I will give an overview of the methods used in the context of the TAC  presenting some general theorems and applications.

Link to slides

Jack Thorne, Symmetric power functoriality for GL(2)
(University of Cambridge)


Thursday, October 20, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 21, 2022 (2am AEDT, 4am NZST)

Abstract: Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2) – the associated symmetric power L-functions are then the ones identified by Serre in the 1960’s in relation to the Sato—Tate conjecture.

I will describe the background to these ideas and then discuss the proof, joint with James Newton, of the existence of these symmetric power liftings for Hilbert modular forms. One arithmetic consequence is that if E is a (non-CM) elliptic curve over a real quadratic field, then all of its symmetric power L-functions admit analytic continuation to the whole complex plane.

Link to slides

Thomas Gauthier, A complex analytic approach to sparsity, rigidity and uniformity in arithmetic dynamics
(Université Paris-Saclay)


Thursday, October 13, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 14, 2022 (2am AEDT, 4am NZST)

Abstract: This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, I will give a strategy to tackle several of those problems at the same time. This is based on a joint work in progress with Gabriel Vigny and Johan Taflin.

Link to slides

Jeffrey C. Lagarias, The Alternative Hypothesis and Point Processes
(University of Michigan)


Thursday, October 6, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 7, 2022 (2am AEDT, 4am NZST)

Abstract: The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that  nearly all  normalized zero spacings be  near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to  the long-standing problem of proving there are no  exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing  Alternative Hypothesis type statistics, which is  consistent  with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by  Tao with slightly different methods.) The talk  reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can  mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.

Link to slides

Paul Nelson, The sup norm problem in the level aspect
(Aarhus University)


Thursday, September 29, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 30, 2022 (1am AEST, 4am NZDT)

Abstract: The sup norm problem concerns the size of L^2-normalized eigenfunctions of manifolds.  In many situations, one expects to be able to improve upon the general bound following from local considerations.  The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface.  Their method has since been extended and applied by many authors, notably to the "level aspect" variant of the problem, where one varies the underlying manifold rather than the eigenvalue.  Recently, Raphael Steiner introduced a new method for attacking the sup norm problem.  I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.

Link to slides

Alexandra Florea, Negative moments of the Riemann zeta function
(University of California Irvine)


Thursday, September 22, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 23, 2022 (1am AEST, 3am NZST)

Abstract: I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.

Link to slides

Danny Neftin, Reducible fibers of polynomial maps
(Technion-Israel Institute of Technology)


Thursday, September 15, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 16, 2022 (1am AEST, 3am NZST)

Abstract: For a polynomial $f\in \mathbb Q[x]$, the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a ``thin" set of exceptions $R_f$ whose explicit description is unknown in general. The problem of describing $R_f$ is closely related to reducibility and arboreal representations in arithmetic dynamics, as well as to Kronecker and arithmetic equivalence for polynomial maps, that is, polynomial versions of the question: "can you hear the shape of the drum?". We shall discuss recent progress on the above problem and topics.

Link to slides

Ping Xi, Analytic approaches towards Katz’s problems on Kloosterman sums
(Xi'an Jiaotong University)


Thursday, September 8, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 9, 2022 (1am AEST, 3am NZST)

Abstract: Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from $\ell$-adic cohomology.

Link to slides

Yann Bugeaud, B'
(University of Strasbourg)


Thursday, September 1, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 2, 2022 (1am AEST, 3am NZST)

Abstract: Link

Link to slides

Alexander Gamburd, Arithmetic and dynamics on varieties of Markoff type
(CUNY Graduate Center)


Thursday, June 30, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 1, 2022 (1am AEST, 3am NZST)

Abstract: The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as  ensuing implications, diophantine and dynamical.


Amir Shpilka, Points, lines and polynomial identities
(Tel Aviv University)


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.

Link to slides

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.

Link to slides

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.

Link to slides

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)