Previous Talks

Mark Shusterman, Counting Minimally Ramified Global Field Extensions
(Harvard University)

Thursday, May 18, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 19, 2023 (1am AEST, 3am NZST)

Abstract: Given a finite group G, one is interested in the number of Galois extensions of a global field with Galois group G and bounded discriminant. We consider a refinement of this problem where the discriminant is required to have the smallest possible number of (distinct) prime factors. We will discuss existing results and conjectures over number fields, and present some recent results over function fields. 

Link to slides

Ben Green, On Sarkozy's theorem for shifted primes
(University of Oxford)

Thursday, May 11, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 12, 2023 (1am AEST, 3am NZST)

Abstract: Suppose that N is large and that A is a subset of {1,..,N} which does not contain two elements x, y with x - y equal to p-1, p a prime. Then A has cardinality at most N^{1 - c}, for some absolute c > 0. I will discuss the history of this kind of question as well as some aspects of the proof of the stated result. 

Link to recording (YouTube)

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Peter Koymans, Counting nilpotent extensions
(University of Michigan)

Thursday, May 4, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 5, 2023 (1am AEST, 3am NZST)

Abstract: We discuss some recent progress towards the strong form of Malle’s conjecture. Even for nilpotent extensions, only very few cases of this conjecture are currently known. We show how equidistribution of Frobenius elements plays an essential role in this problem and how this can be used to make further progress towards Malle’s conjecture. We will also discuss applications to the Massey vanishing conjecture and to lifting problems. This is joint work with Carlo Pagano.

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Pierre Le Boudec, 2-torsion in class groups of number fields
(University of Basel)

Thursday, April 27, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 28, 2023 (1am AEST, 3am NZST)

Abstract: It is well-known that the class number of a number field K of fixed degree n is roughly bounded by the square root of the absolute value of the discriminant of K. However, given a prime number p, the cardinality of the p-torsion subgroup of the class group of K is expected to be much smaller. Unfortunately, beating the trivial bound mentioned above is a hard problem. Indeed, this task had only been achieved for a handful of pairs (n,p) until Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao managed to do so for any degree n in the case p=2. In this talk we will go through their proof and we will present new bounds which depend on the geometry of the lattice underlying the ring of integers of K. This is joint work with Dante Bonolis. 

Link to recording (YouTube)

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Bjorn Poonen, Integral points on curves via Baker's method and finite étale covers
(MIT)

Thursday, April 20, 2023 (6am PDT, 9am EDT, 2pm GMT, 3pm CEST, 4pm Israel Daylight Time, 6:30pm Indian Standard Time, 9pm CST, 11pm AEST)
Friday, April 21, 2023 (1am NZST)

Abstract: We prove results in the direction of showing that for some affine curves, Baker's method applied to finite étale covers is insufficient to determine the integral points.

Link to recording (YouTube)

Hélène Esnault, Integrality Properties of the Betti Moduli Space
(Freie Universität Berlin)

Thursday, April 13, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 14, 2023 (1am AEST, 3am NZST)

Abstract: We use de Jong’s conjecture and the existence of $\ell$-adic companions to single out integrality properties of the Betti moduli space. The first such instance was in joint work with Michael Groechenig on Simpson’s integrality conjecture for (cohomologically) rigid local systems. This integrality property yields an obstruction for a finitely presented group to be the fundamental group of a sooth quasi-projective complex variety. (joint with Johan de Jong)

Link to recording (YouTube)

Kannan Soundararajan, Covering integers using quadratic forms
(Stanford University)

Thursday, April 6, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 7, 2023 (1am AEST, 3am NZST)

Abstract: How large must $\Delta$ be so that we can cover a substantial proportion of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ with $d$ below $\Delta$?  Problems involving representations by binary quadratic forms have a long history, going back to Fermat.  The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao.  In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition:  If $\Delta$ is below $(\log X)^{\log 2-\epsilon}$ then zero percent of the integers below $X$ are represented, whereas if $\Delta$ is above $(\log X)^{\log 2 +\epsilon}$ then 100 percent of the integers below $X$ are represented.  

Link to recording

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Ziyang Gao, Sparsity of rational and algebraic points
(Leibniz University Hannover)

Thursday, March 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, March 31, 2023 (2am AEDT, 4am NZDT)

Abstract: It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.

In my talk, I will explain the historical and recent developments of this problem according to the different grades.

Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.

If time permits, I will mention the corresponding result in high dimensions.

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Ofir Gorodetsky, How many smooth numbers and smooth polynomials are there?
(University of Oxford)

Thursday, March 23, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 24, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields.

We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)^2 (as conjectured by Hildebrand) and y=(log x)^(3/2), when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.

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Florian Luca, Recent progress on the Skolem problem
(University of the Witwatersrand)

Thursday, March 16, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 17, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. One of the main contributions of the talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method. 

 In the second part of the talk, we present the notion of an Universal Skolem Set, which is a subset of the positive integers on which the Skolem is decidable regardless of the linear recurrence. We give two examples of such sets, one of which is of positive density (that is, contains a positive proportion of all the positive integers).

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Alex Wilkie, Integer points on analytic sets
(University of Oxford and University of Manchester)

Thursday, March 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 10, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2004 I proved an O(loglogH) bound for the number of integer points of height at most H lying on a globally subanalytic curve. (The paper was published in the Journal of Symbolic Logic and so probably escaped the notice of most of you reading this.) Recently, Gareth Jones and Gal Binyamini proposed a generalization of the result to higher dimensions (where the obvious statement is almost certainly false) and I shall report on our joint work: one obtains the (hoped for) (loglogH)^n  bound for (not globally subanalytic but) globally analytic sets of dimension n.  

Link to recording

Alexandra Shlapentokh, Defining integers using unit groups
(East Carolina University)

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

Abstract: We discuss some problems of definability and decidability over rings of integers of algebraic extensions of $\Q$.  In particular, we show that for a large class of fields $K$ there is a simple formula defining rational integers over $O_K$.  Below $U_K$ is the group of units of $O_K$. 

$\Z=\{x| \forall \varepsilon \in U_K\setminus \{1\}\ \exists \delta \in U_K: x \equiv \frac{\delta-1}{\varepsilon-1} \bmod (\varepsilon-1)\}$. This talk is based on a joint paper with Barry Mazur and Karl Rubin.

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Terence Tao, Infinite Partial Sumsets in the Primes
(UCLA)

Thursday, February 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 24, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture).  Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 < a_2 < ... and b_1 < b_2 < ... such that a_i + b_j is prime for all i<j.  Equivalently, the primes are not "translation-finite" in the sense of Ruppert.  As an application of these methods we show that the orbit closure of the primes is uncountable.

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Wanlin Li, Ordinary and Basic Reductions of Abelian Varieties
(Université de Montréal)

Thursday, February 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 17, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc). 

In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.

Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.

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Sarah Peluse, Divisibility of character values of the symmetric group
(Institute for Advanced Study and Princeton University)

Thursday, February 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 10, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2017, Miller computed the character tables of $S_n$ for all $n$ up to $38$ and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of $S_n$ are divisible by any fixed prime power as $n$ tends to infinity. In this talk, I will discuss joint work with K. Soundararajan that resolves this conjecture, and mention some related open problems.

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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.

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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.

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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.

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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.

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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.

Link to recording (294MB)

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.

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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.

Link to recording (124MB)

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 recording (140MB)

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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 1986 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.

Link to recording (94MB)

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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.

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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.

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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.

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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.

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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.

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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.

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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 recording (96MB)

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 recording (118MB)

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 recording (110MB)

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 recording (166MB)

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 recording (111MB)

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 recording (283MB)

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 recording (132MB)

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 recording (112MB)

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.

Link to recording (323MB)

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 recording (116MB)

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 recording (136MB)

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 recording (101MB)

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.

Link to recording (121MB)

Link to slides and slides with annotations

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 recording (116MB)

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 recording (109MB)

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 recording (108MB)

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 recording (109MB)

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 recording (124MB)

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 recording (111MB)

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 recording (85MB)

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 recording (163MB)

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 recording (104MB)

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 recording (150MB)

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 recording (131MB)

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 recording (111MB)

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)

Link to recording (103MB)

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 recording (149MB)

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.

Link to recording (99MB)

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 recording (124MB)

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 recording (129MB)

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.

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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).

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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.

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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.

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Johan Commelin, Liquid Tensor Experiment
(Albert–Ludwigs-Universität Freiburg)