# Previous Talks

## Jozsef Solymosi, *Rank of matrices with entries from a multiplicative group*

(University of British Columbia)

Jozsef Solymosi,

*Rank of matrices with entries from a multiplicative group*

(University of British Columbia)

**Thursday, ****January 20****, 202****2** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****January 21****, 202****2** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**January 20**

**, 202**

**2**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**January 21**

**, 202**

**2**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets. (Joint work with Noga Alon)

*Abstract:*We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets. (Joint work with Noga Alon)

## Péter Varjú, *Irreducibility of random polynomials*

(University of Cambridge)

Péter Varjú,

*Irreducibility of random polynomials*

(University of Cambridge)

**Thursday, ****January 13****, 202****2** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****January 14****, 202****2** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**January 13**

**, 202**

**2**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**January 14**

**, 202**

**2**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* Consider random polynomials of degree d whose leading and constant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts that such a polynomial is irreducible in Z[x] with high probability as d grows. This conjecture is still open, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the method of proof of this result and will discuss later developments that apply this method to other models of random polynomials.

*Abstract:*Consider random polynomials of degree d whose leading and constant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts that such a polynomial is irreducible in Z[x] with high probability as d grows. This conjecture is still open, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the method of proof of this result and will discuss later developments that apply this method to other models of random polynomials.

## Sarah Zerbes, *Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces*

(University College London, UK)

Sarah Zerbes,

*Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces*

(University College London, UK)

**Thursday, ****December 16****, 202****1** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****December 17****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**December 16**

**, 202**

**1**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**December 17**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* Euler systems are one of the most powerful tools for proving cases of the Bloch--Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture.

*Abstract:*Euler systems are one of the most powerful tools for proving cases of the Bloch--Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture.

### I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.

I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.

## Samir Siksek, *The Fermat equation and the unit equation*

(University of Warwick)

Samir Siksek,

*The Fermat equation and the unit equation*

(University of Warwick)

**Thursday, ****December 9****, 202****1** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****December 10****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**December 9**

**, 202**

**1**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**December 10**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* The asymptotic Fermat conjecture (AFC) states that for a number field K, and for sufficiently large primes p, the only solutions to the Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch recent work that connects the Fermat equation to the far more elementary unit equation, and explain how this surprising connection can be exploited to prove AFC for several infinite families of number fields. This talk is based on joint work with Nuno Freitas, Alain Kraus and Haluk Sengun.

*Abstract:*The asymptotic Fermat conjecture (AFC) states that for a number field K, and for sufficiently large primes p, the only solutions to the Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch recent work that connects the Fermat equation to the far more elementary unit equation, and explain how this surprising connection can be exploited to prove AFC for several infinite families of number fields. This talk is based on joint work with Nuno Freitas, Alain Kraus and Haluk Sengun.

## Kiran Kedlaya, *Orders of abelian varieties over $\mathbb{F}_2$*

(University of California San Diego)

Kiran Kedlaya,

*Orders of abelian varieties over $\mathbb{F}_2$*

(University of California San Diego)

**Thursday, ****December 2****, 202****1** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****December 3****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**December 2**

**, 202**

**1**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**December 3**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.

*Abstract:*We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.

## Alexei Skorobogatov, *On uniformity conjectures for abelian varieties and K3 surfaces*

(Imperial College London)

Alexei Skorobogatov,

*On uniformity conjectures for abelian varieties and K3 surfaces*

(Imperial College London)

**Thursday, ****November 25****, 202****1** *(8am P**S**T, 11am E**S**T, 4pm **GMT**, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****November 26****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**November 25**

**, 202**

**1**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm*

*GMT*

*, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**November 26**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* I will discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Néron–Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group. The talk is based on a joint work with Martin Orr and Yuri Zarhin.

*Abstract:*I will discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Néron–Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group. The talk is based on a joint work with Martin Orr and Yuri Zarhin.

## Myrto Mavraki, *Towards uniformity in the dynamical Bogomolov conjecture*

(Harvard University)

Myrto Mavraki,

*Towards uniformity in the dynamical Bogomolov conjecture*

(Harvard University)

**Thursday, ****November 18****, 202****1** *(8am P**S**T, 11am E**S**T, 4pm BST, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****November 19****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**November 18**

**, 202**

**1**

*(8am P*

*S*

*T, 11am E*

*S*

*T, 4pm BST, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**November 19**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.

*Abstract:*Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.

### In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end, we establish an instance of a 'relative dynamical Bogomolov'. This is work in progress joint with Harry Schmidt (University of Basel).

In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end, we establish an instance of a 'relative dynamical Bogomolov'. This is work in progress joint with Harry Schmidt (University of Basel).

## Avi Wigderson, *Randomness*

(Institute for Advanced Study)

Avi Wigderson,

*Randomness*

(Institute for Advanced Study)

**Thursday, ****November 11****, 202****1 ***(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)*

**Friday, ****November 12****, 202****1** *(**12am CST, 3am AEDT, 5am NZDT)*

**Thursday,**

**November 11**

**, 202**

**1**

*(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)*

**Friday,**

**November 12**

**, 202**

**1**

*(*

*12am CST, 3am AEDT, 5am NZDT)*

*Abstract:* Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference between the two?

*Abstract:*Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference between the two?

### Humanity has pondered the meaning and utility of randomness for millennia.

Humanity has pondered the meaning and utility of randomness for millennia.

### There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable! Which of these applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from somewhat "unpredictable" phenomena like the weather or the stock market?

There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable! Which of these applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is available, e.g. that arises from somewhat "unpredictable" phenomena like the weather or the stock market?

### A computational theory of randomness, developed in the past several decades, reveals (perhaps counter-intuitively) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory, notions of pseudo-randomness, and connections to computational intractability.

A computational theory of randomness, developed in the past several decades, reveals (perhaps counter-intuitively) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory, notions of pseudo-randomness, and connections to computational intractability.

### It is interesting that Number Theory played an important role throughout this development. It supplied problems whose algorithmic solution make randomness seem powerful, problems for which randomness can be eliminated from such solutions, and problems where the power of randomness remains a major challenge for computational complexity theorists and mathematicians. I will use these problems (and others) to demonstrate aspects of this theory.

It is interesting that Number Theory played an important role throughout this development. It supplied problems whose algorithmic solution make randomness seem powerful, problems for which randomness can be eliminated from such solutions, and problems where the power of randomness remains a major challenge for computational complexity theorists and mathematicians. I will use these problems (and others) to demonstrate aspects of this theory.

## Katherine Stange, *Algebraic Number Starscapes*

(University of Colorado, Boulder)

Katherine Stange,

*Algebraic Number Starscapes*

(University of Colorado, Boulder)

**Thursday, ****November 4****, 202****1** *(**9**am P**D**T, 1**2p**m E**D**T, 4pm BST, 5pm CET, 6pm **Israel Daylight Time**, **9**:30pm Indian Standard Time)*

**Friday, ****November 5****, 202****1** *(1**2a**m CST, **3am AEDT, 5am NZDT**)*

**Thursday,**

**November 4**

**, 202**

**1**

*(*

*9*

*am P*

*D*

*T, 1*

*2p*

*m E*

*D*

*T, 4pm BST, 5pm CET, 6pm*

*Israel Daylight Time*

*,*

*9*

*:30pm Indian Standard Time)*

**Friday,**

**November 5**

**, 202**

**1**

*(1*

*2a*

*m CST,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* In the spirit of experimentation, at the Fall 2019 ICERM special semester on “Illustrating Mathematics,” I began drawing algebraic numbers in the complex plane. Edmund Harriss, Steve Trettel and I sized the numbers by arithmetic complexity and found a wealth of pattern and structure. In this talk, I’ll take you on a visual tour and share some of the mathematical explanations we found for what can be quite stunning pictures (in the hands of a mathematician and artist like Edmund). This experience gave me a new perspective on complex Diophantine approximation: one can view approximation properties as being dictated by the geometry of the map from coefficient space to root space in different polynomial degrees. I’ll explain this geometry, and discuss a few Diophantine results, known and new, in this context.

*Abstract:*In the spirit of experimentation, at the Fall 2019 ICERM special semester on “Illustrating Mathematics,” I began drawing algebraic numbers in the complex plane. Edmund Harriss, Steve Trettel and I sized the numbers by arithmetic complexity and found a wealth of pattern and structure. In this talk, I’ll take you on a visual tour and share some of the mathematical explanations we found for what can be quite stunning pictures (in the hands of a mathematician and artist like Edmund). This experience gave me a new perspective on complex Diophantine approximation: one can view approximation properties as being dictated by the geometry of the map from coefficient space to root space in different polynomial degrees. I’ll explain this geometry, and discuss a few Diophantine results, known and new, in this context.

### Link to website

Link to website

## Dimitris Koukoulopoulos, *Towards a high-dimensional theory of divisors of integers*

(University of Montreal)

Dimitris Koukoulopoulos,

*Towards a high-dimensional theory of divisors of integers*

(University of Montreal)

**Thursday, ****October 28****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****October 29****, 202****1** *(**2am AEDT, 4am NZDT**)*

**Thursday,**

**October 28**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**October 29**

**, 202**

**1**

*(*

*2am AEDT, 4am NZDT*

*)*

*Abstract:* In this talk, I will survey some results about high-dimensional phenomena in the theory of divisors of integers.

*Abstract:*In this talk, I will survey some results about high-dimensional phenomena in the theory of divisors of integers.

### Fix an integer $k\ge2$ and pick an integer $n\le x$ uniformly at random. We then consider the following two basic problems:

Fix an integer $k\ge2$ and pick an integer $n\le x$ uniformly at random. We then consider the following two basic problems:

### What are the chances that $n$ can be factored as $n=d_1\cdots d_k$ with each factor $d_i$ lying in some prescribed dyadic interval $[y_i,2y_i]$?

What are the chances that $n$ can be factored as $n=d_1\cdots d_k$ with each factor $d_i$ lying in some prescribed dyadic interval $[y_i,2y_i]$?

### What are the chances that we can find $k$ divisors of $n$, say $d_1,\dots,d_k$, such that $|\log(d_j/d_i)|<1$ for all $i,j$, and which are all composed from a prescribed set of prime factors of $n$?

What are the chances that we can find $k$ divisors of $n$, say $d_1,\dots,d_k$, such that $|\log(d_j/d_i)|<1$ for all $i,j$, and which are all composed from a prescribed set of prime factors of $n$?

### The first problem is a high-dimensional generalization of the Erd\H os multiplication table problem; it is well-understood when $k\le 6$, but less so when $k\ge7$. The second problem is related to Hooley’s function $\Delta(n):=\max_u \#\{d|n:u<\log d\le u+1\}$ that measures the concentration of the sequence of divisors of $n$, and that has surprising applications to Diophantine number theory.

The first problem is a high-dimensional generalization of the Erd\H os multiplication table problem; it is well-understood when $k\le 6$, but less so when $k\ge7$. The second problem is related to Hooley’s function $\Delta(n):=\max_u \#\{d|n:u<\log d\le u+1\}$ that measures the concentration of the sequence of divisors of $n$, and that has surprising applications to Diophantine number theory.

### In recent work with Kevin Ford and Ben Green, we built on the earlier work on Problem 1 to develop a new approach to Problem 2. This led to an improved lower bound on the almost-sure behaviour of Hooley’s $\Delta$-function, that we conjecture to be optimal. The new ideas might in turn shed light to Problem 1 and other high-dimensional phenomena about divisors of integers.

In recent work with Kevin Ford and Ben Green, we built on the earlier work on Problem 1 to develop a new approach to Problem 2. This led to an improved lower bound on the almost-sure behaviour of Hooley’s $\Delta$-function, that we conjecture to be optimal. The new ideas might in turn shed light to Problem 1 and other high-dimensional phenomena about divisors of integers.

## Johan Commelin, *Liquid Tensor Experiment*

(Albert–Ludwigs-Universität Freiburg)

Johan Commelin,

*Liquid Tensor Experiment*

(Albert–Ludwigs-Universität Freiburg)

**Thursday, ****October 21****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****October 22****, 202****1** *(**2am AEST, 4am NZST**)*

**Thursday,**

**October 21**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**October 22**

**, 202**

**1**

*(*

*2am AEST, 4am NZST*

*)*

*Abstract:* In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid $\mathbb{R}$-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that in a couple of months we will have completed the full challenge.

*Abstract:*In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid $\mathbb{R}$-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that in a couple of months we will have completed the full challenge.

### In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.

In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.

### Link to public chat log (txt file)

Link to public chat log (txt file)

## Jean-Marc Deshouillers, *Are factorials sums of three cubes?*

(Institut de Mathématiques de Bordeaux)

Jean-Marc Deshouillers,

*Are factorials sums of three cubes?*

(Institut de Mathématiques de Bordeaux)

**Thursday, ****October 14****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****October 15****, 202****1** *(**2am AEST, 4am NZST**)*

**Thursday,**

**October 14**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**October 15**

**, 202**

**1**

*(*

*2am AEST, 4am NZST*

*)*

## Henryk Iwaniec, *Remarks on the large sieve*

(Rutgers University)

Henryk Iwaniec,

*Remarks on the large sieve*

(Rutgers University)

**A talk in honor of John Friedlander's 80th birthday**

**Special Chairs: Leo Goldmakher (Williams College) and Andrew Granville (University of Montreal)**

**Special Chairs: Leo Goldmakher (Williams College) and Andrew Granville (University of Montreal)**

**Thursday, ****October 7****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****October 8****, 202****1** *(**2am AEST, 4am NZST**)*

**Thursday,**

**October 7**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**October 8**

**, 202**

**1**

*(*

*2am AEST, 4am NZST*

*)*

*Abstract:* The concept of the large sieve will be discussed in various contexts. The power and limitation of basic estimates will be illustrated with some examples. Recent work on the large sieve for characters to prime moduli will be explained.

*Abstract:*The concept of the large sieve will be discussed in various contexts. The power and limitation of basic estimates will be illustrated with some examples. Recent work on the large sieve for characters to prime moduli will be explained.

### Link to slides

Link to slides

## Anish Ghosh, *Values of quadratic forms at integer points*

(Tata Institute of Fundamental Research)

Anish Ghosh,

*Values of quadratic forms at integer points*

(Tata Institute of Fundamental Research)

**Thursday, ****September 30****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****October 1****, 202****1** *(**1am AEST, 4am NZDT**)*

**Thursday,**

**September 30**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**October 1**

**, 202**

**1**

*(*

*1am AEST, 4am NZDT*

*)*

*Abstract:* A famous theorem of Margulis, resolving a conjecture of Oppenheim, states that an indefinite, irrational quadratic form in at least three variables takes a dense set of values at integer points. Recently there has been a push towards establishing effective versions of Margulis's theorem. I will explain Margulis's approach to this problem which involves the ergodic theory of group actions on homogeneous spaces. I will then discuss some new effective results in this direction. These results use a variety of techniques including tools from ergodic theory, analytic number theory as well as the geometry of numbers.

*Abstract:*A famous theorem of Margulis, resolving a conjecture of Oppenheim, states that an indefinite, irrational quadratic form in at least three variables takes a dense set of values at integer points. Recently there has been a push towards establishing effective versions of Margulis's theorem. I will explain Margulis's approach to this problem which involves the ergodic theory of group actions on homogeneous spaces. I will then discuss some new effective results in this direction. These results use a variety of techniques including tools from ergodic theory, analytic number theory as well as the geometry of numbers.

## Alina Carmen Cojocaru, *Bounds for the distribution of the Frobenius traces associated to abelian varieties*

(University of Illinois at Chicago and Institute of Mathematics of the Romanian Academy)

Alina Carmen Cojocaru,

*Bounds for the distribution of the Frobenius traces associated to abelian varieties*

(University of Illinois at Chicago and Institute of Mathematics of the Romanian Academy)

**Thursday, ****September 23****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****September 24****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**September 23**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**September 24**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* In 1976, Serge Lang and Hale Trotter conjectured the asymptotic growth of the number $\pi_A(x, t)$ of primes $p < x$ for which the Frobenius trace $a_p$ of a non-CM elliptic curve $A/\mathbb{Q}$ equals an integer $t$. Even though their conjecture remains open, over the past decades the study of the counting function $\pi_A(x, t)$ has witnessed remarkable advances. We will discuss generalizations of such studies in the setting of an abelian variety $A/\mathbb{Q}$ of arbitrary dimension and we will present non-trivial upper bounds for the corresponding counting function $\pi_A(x, t)$. This is joint work with Tian Wang (University of Illinois at Chicago).

*Abstract:*In 1976, Serge Lang and Hale Trotter conjectured the asymptotic growth of the number $\pi_A(x, t)$ of primes $p < x$ for which the Frobenius trace $a_p$ of a non-CM elliptic curve $A/\mathbb{Q}$ equals an integer $t$. Even though their conjecture remains open, over the past decades the study of the counting function $\pi_A(x, t)$ has witnessed remarkable advances. We will discuss generalizations of such studies in the setting of an abelian variety $A/\mathbb{Q}$ of arbitrary dimension and we will present non-trivial upper bounds for the corresponding counting function $\pi_A(x, t)$. This is joint work with Tian Wang (University of Illinois at Chicago).

## Martín Sombra, *The mean height of the solution set of a system of polynomial equations*

(ICREA and University of Barcelona)

Martín Sombra,

*The mean height of the solution set of a system of polynomial equations*

(ICREA and University of Barcelona)

**Thursday, ****September 16****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****September 17****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**September 16**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**September 17**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Bernstein’s theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, we can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.

*Abstract:*Bernstein’s theorem allows to predict the number of solutions of a system of Laurent polynomial equations in terms of combinatorial invariants. When the coefficients of the system are algebraic numbers, we can ask about the height of these solutions. Based on an on-going project with Roberto Gualdi (Regensburg), I will explain how one can approach this question using tools from the Arakelov geometry of toric varieties.

## Emmanuel Kowalski, *Harmonic analysis over finite fields and equidistribution*

(ETH Zürich)

Emmanuel Kowalski,

*Harmonic analysis over finite fields and equidistribution*

(ETH Zürich)

**Thursday, ****September 9****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****September 10****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**September 9**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**September 10**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* In 1976, Deligne defined a geometric version of the Fourier transform over finite fields, leading to significant applications in number theory.

*Abstract:*In 1976, Deligne defined a geometric version of the Fourier transform over finite fields, leading to significant applications in number theory.

### For a number of applications, including equidistribution of exponential sums parameterized by multiplicative characters, it would be very helpful to have a similar geometric harmonic analysis for other groups. I will discuss ongoing joint work with A. Forey and J. Fresán in which we establish some results in this direction by generalizing ideas of Katz. I will present the general equidistribution theorem for exponential sums parameterized by characters that we obtain, and discuss applications, as well as challenges, open questions and mysteries.

For a number of applications, including equidistribution of exponential sums parameterized by multiplicative characters, it would be very helpful to have a similar geometric harmonic analysis for other groups. I will discuss ongoing joint work with A. Forey and J. Fresán in which we establish some results in this direction by generalizing ideas of Katz. I will present the general equidistribution theorem for exponential sums parameterized by characters that we obtain, and discuss applications, as well as challenges, open questions and mysteries.

## Lars Kühne, *The uniform Bogomolov conjecture for algebraic curves*

(University of Copenhagen)

Lars Kühne,

*The uniform Bogomolov conjecture for algebraic curves*

(University of Copenhagen)

**Thursday, ****September 2****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****September 3****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**September 2**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**September 3**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces, but it also follows from independent work by Yuan and Zhang, which has been recently reported in this seminar. I will therefore focus on the application that motivated my work, namely a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell-Lang conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will restrict to the case of curves for simplicity.

*Abstract:*I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces, but it also follows from independent work by Yuan and Zhang, which has been recently reported in this seminar. I will therefore focus on the application that motivated my work, namely a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell-Lang conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will restrict to the case of curves for simplicity.

## Alexandru Zaharescu, *Some remarks on Landau - Siegel zeros*

(University of Illinois at Urbana-Champaign)

Alexandru Zaharescu,

*Some remarks on Landau - Siegel zeros*

(University of Illinois at Urbana-Champaign)

**Thursday, ****August 26****, 202****1** *(**2**pm PDT, 5pm EDT, 10pm BST, 11pm CEST**)*

**Friday, ****August 27****, 202****1** *(**12am Israel Daylight Time, 2:30am Indian Standard Time, 5**am CST, 7am AEST, 9am NZST**)*

**Thursday,**

**August 26**

**, 202**

**1**

*(*

*2*

*pm PDT, 5pm EDT, 10pm BST, 11pm CEST*

*)*

**Friday,**

**August 27**

**, 202**

**1**

*(*

*12am Israel Daylight Time, 2:30am Indian Standard Time, 5*

*am CST, 7am AEST, 9am NZST*

*)*

*Abstract:* In the first part of the talk I will survey some known results related to the hypothetical existence of Landau - Siegel zeros. In the second part of the talk I will discuss some recent joint work with Hung Bui and Kyle Pratt in which we show that the existence of Landau - Siegel zeros has implications for the behavior of L - functions at the central point.

*Abstract:*In the first part of the talk I will survey some known results related to the hypothetical existence of Landau - Siegel zeros. In the second part of the talk I will discuss some recent joint work with Hung Bui and Kyle Pratt in which we show that the existence of Landau - Siegel zeros has implications for the behavior of L - functions at the central point.

## Zeev Dvir, *The Kakeya set conjecture over rings of integers modulo square free m*

(Princeton University)

Zeev Dvir,

*The Kakeya set conjecture over rings of integers modulo square free m*

(Princeton University)

**Thursday, ****August 19****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****August 20****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**August 19**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**August 20**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* We show that, when N is any square-free integer, the size of the smallest Kakeya set in (ℤ/Nℤ)^n is at least C_{eps,n}*N^{n-eps} for any eps>0 -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the p-rank of the incidence matrix of points and hyperplanes over (ℤ/p^kℤ)^n. Joint work with Manik Dhar.

*Abstract:*We show that, when N is any square-free integer, the size of the smallest Kakeya set in (ℤ/Nℤ)^n is at least C_{eps,n}*N^{n-eps} for any eps>0 -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime N. We also show that the case of general N can be reduced to lower bounding the p-rank of the incidence matrix of points and hyperplanes over (ℤ/p^kℤ)^n. Joint work with Manik Dhar.

## Francesco Amoroso, *Bounded Height in Pencils of Subgroups of finite rank*

(University of Caen)

Francesco Amoroso,

*Bounded Height in Pencils of Subgroups of finite rank*

(University of Caen)

**Thursday, ****August 12****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****August 13****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**August 12**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**August 13**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* [Joint work with D.Masser and U.Zannier]

*Abstract:*[Joint work with D.Masser and U.Zannier]

### Let n>1 be a varying natural number. By a result of Beukers, the solutions of t^n+(1-t)^n=1 have uniformly bounded height. What happens if we allow rational exponents?

Let n>1 be a varying natural number. By a result of Beukers, the solutions of t^n+(1-t)^n=1 have uniformly bounded height. What happens if we allow rational exponents?

### We consider the analogous question replacing the affine curve x+y=1 with an arbitrary irreducible curve and {t^n | n rational} with the division group of a finitely generated subgroup.

We consider the analogous question replacing the affine curve x+y=1 with an arbitrary irreducible curve and {t^n | n rational} with the division group of a finitely generated subgroup.

## Frank Calegari, *Digits*

(University of Chicago)

Frank Calegari,

*Digits*

(University of Chicago)

**Thursday, ****August 5****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****August 6****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**August 5**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**August 6**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This is work in progress with Soundararajan which we may well write up one day.

*Abstract:*We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This is work in progress with Soundararajan which we may well write up one day.

## Arno Fehm, *Is Z diophantine in Q?*

(Technische Universität Dresden)

Arno Fehm,

*Is Z diophantine in Q?*

(Technische Universität Dresden)

**Thursday, ****July 29****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****July 30****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**July 29**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**July 30**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Are the integers the projection of the rational zeros of a polynomial in several variables onto the first coordinate? The aim of this talk is to motivate and discuss this longstanding question. I will survey some results regarding diophantine sets and Hilbert's tenth problem (the existence of an algorithm that decides whether a polynomial has a zero) in fields and will discuss a few conjectures, some classical and some more recent, that suggest that the answer to the question should be negative.

*Abstract:*Are the integers the projection of the rational zeros of a polynomial in several variables onto the first coordinate? The aim of this talk is to motivate and discuss this longstanding question. I will survey some results regarding diophantine sets and Hilbert's tenth problem (the existence of an algorithm that decides whether a polynomial has a zero) in fields and will discuss a few conjectures, some classical and some more recent, that suggest that the answer to the question should be negative.

## Kumar Murty, *Periods and Mixed Motives*

(University of Toronto)

Kumar Murty,

*Periods and Mixed Motives*

(University of Toronto)

**Thursday, ****July 22****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****July 23****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**July 22**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**July 23**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* We discuss some consequences of Grothendieck's Period Conjecture in the context of mixed motives. In particular, this conjecture implies that zeta(3), log 2 and pi are algebraically independent (contrary to an expectation of Euler). After some 'motivation' and introductory remarks on periods, we derive our consequences as a result of studying mixed motives whose Galois group has a large unipotent radical. This is joint work with Payman Eskandari.

*Abstract:*We discuss some consequences of Grothendieck's Period Conjecture in the context of mixed motives. In particular, this conjecture implies that zeta(3), log 2 and pi are algebraically independent (contrary to an expectation of Euler). After some 'motivation' and introductory remarks on periods, we derive our consequences as a result of studying mixed motives whose Galois group has a large unipotent radical. This is joint work with Payman Eskandari.

## Ricardo Menares, *p-adic distribution of CM points*

(Pontificia Universidad Católica de Chile)

Ricardo Menares,

*p-adic distribution of CM points*

(Pontificia Universidad Católica de Chile)

**Thursday, ****July 15****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****July 16****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**July 15**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**July 16**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* CM points are the isomorphism classes of CM elliptic curves. When ordered by the absolute value of the discriminant of the endomorphism ring, CM points are distributed along the complex (level one) modular curve according to the hyperbolic measure. This statement was proved by Duke for fundamental discriminants and later, building on this work, Clozel and Ullmo proved it in full generality.

*Abstract:*CM points are the isomorphism classes of CM elliptic curves. When ordered by the absolute value of the discriminant of the endomorphism ring, CM points are distributed along the complex (level one) modular curve according to the hyperbolic measure. This statement was proved by Duke for fundamental discriminants and later, building on this work, Clozel and Ullmo proved it in full generality.

### In this talk, we establish the p-adic analogue of this result. Namely, for a fixed prime p we regard the CM points as a subset of the p-adic space attached to the modular curve and we classify the possible accumulation measures of CM points as the discriminant varies. In particular, we find that there are infinitely many such measures. This is in stark contrast to the complex case, where the hyperbolic measure is the unique accumulation measure.

In this talk, we establish the p-adic analogue of this result. Namely, for a fixed prime p we regard the CM points as a subset of the p-adic space attached to the modular curve and we classify the possible accumulation measures of CM points as the discriminant varies. In particular, we find that there are infinitely many such measures. This is in stark contrast to the complex case, where the hyperbolic measure is the unique accumulation measure.

### As an application, we show that for any finite set S of prime numbers, the set of singular moduli which are S-units is finite.

As an application, we show that for any finite set S of prime numbers, the set of singular moduli which are S-units is finite.

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

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

## Brian Conrey, *Moments, ratios, arithmetic functions in short intervals and random matrix averages*

(American Institute of Mathematics)

Brian Conrey,

*Moments, ratios, arithmetic functions in short intervals and random matrix averages*

(American Institute of Mathematics)

**Thursday, ****July 8****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****July 9****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**July 8**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**July 9**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* We discuss how the conjectures for moments of L-functions imply short interval averages of the L-coefficient convolutions. Similarly the ratios conjectures lead to short interval averages of the convolutions of coefficients at almost primes. These in turn are related to random matrix averages considered by Diaconis - Gamburd and by Diaconis - Shahshahani.

*Abstract:*We discuss how the conjectures for moments of L-functions imply short interval averages of the L-coefficient convolutions. Similarly the ratios conjectures lead to short interval averages of the convolutions of coefficients at almost primes. These in turn are related to random matrix averages considered by Diaconis - Gamburd and by Diaconis - Shahshahani.

## Manjul Bhargava, *Galois groups of random integer polynomials*

(Princeton University)

Manjul Bhargava,

*Galois groups of random integer polynomials*

(Princeton University)

**A talk in honor of Don Zagier's 70th birthday**

**Special Chair: Pieter Moree (Max Planck Institute for Mathematics)**

**Special Chair: Pieter Moree (Max Planck Institute for Mathematics)**

**Thursday, ****July 1****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****July 2****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**July 1**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**July 2**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we prove the "Weak van der Waerden Conjecture", which states that the number of such polynomials is $O_\epsilon(H^{n-1+\epsilon})$, for all degrees $n$.

*Abstract:*Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we prove the "Weak van der Waerden Conjecture", which states that the number of such polynomials is $O_\epsilon(H^{n-1+\epsilon})$, for all degrees $n$.

## Annette Huber-Klawitter, *Periods and O-minimality *

(University of Freiburg)

Annette Huber-Klawitter,

*Periods and O-minimality*

(University of Freiburg)

**Thursday, ****June 24****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****June 25****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**June 24**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**June 25**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Roughly, periods are numbers obtained by integrating algebraic differential forms over domains of integration also of arithmetic nature. I am going to give a survey on the state of the period conjecture and different points of view. I also want to present a relation to o-minimal geometry.

*Abstract:*Roughly, periods are numbers obtained by integrating algebraic differential forms over domains of integration also of arithmetic nature. I am going to give a survey on the state of the period conjecture and different points of view. I also want to present a relation to o-minimal geometry.

## Shou-Wu Zhang, *Adelic line bundles over quasi-projective varieties *

(Princeton University)

Shou-Wu Zhang,

*Adelic line bundles over quasi-projective varieties*

(Princeton University)

**Thursday, ****June 17****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****June 18****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**June 17**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**June 18**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* For quasi-projective varieties over finitely generated fields, we develop a theory of adelic line bundles including an equidistribution theorem for Galois orbits of small points. In this lecture, we will explain this theory and its application to arithmetic of abelian varieties, dynamical systems, and their moduli. This is a joint work with Xinyi Yuan.

*Abstract:*For quasi-projective varieties over finitely generated fields, we develop a theory of adelic line bundles including an equidistribution theorem for Galois orbits of small points. In this lecture, we will explain this theory and its application to arithmetic of abelian varieties, dynamical systems, and their moduli. This is a joint work with Xinyi Yuan.

## Matthew Young, *The Weyl bound for Dirichlet L-functions *

(Texas A&M University)

Matthew Young,

*The Weyl bound for Dirichlet L-functions*

(Texas A&M University)

**Thursday, ****June 10****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****June 11****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**June 10**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**June 11**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* There is an analogy between the behavior of the Riemann zeta function high in the critical strip, and the behavior of Dirichlet L-functions of large conductors. In many important problems, our understanding of Dirichlet L-functions is weaker than for zeta; for example, the zero-free regions are not of the same quality due to the possible Landau-Siegel zero. This talk will discuss recent progress (joint with Ian Petrow) on subconvexity bounds for Dirichlet L-functions. These new bounds now match the original subconvexity bound for the zeta function derived by Hardy and Littlewood using Weyl's differencing method.

*Abstract:*There is an analogy between the behavior of the Riemann zeta function high in the critical strip, and the behavior of Dirichlet L-functions of large conductors. In many important problems, our understanding of Dirichlet L-functions is weaker than for zeta; for example, the zero-free regions are not of the same quality due to the possible Landau-Siegel zero. This talk will discuss recent progress (joint with Ian Petrow) on subconvexity bounds for Dirichlet L-functions. These new bounds now match the original subconvexity bound for the zeta function derived by Hardy and Littlewood using Weyl's differencing method.

## Antoine Chambert-Loir, *From complex function theory to non-archimedean spaces - a number theoretical thread*

(Université Paris-Diderot)

Antoine Chambert-Loir,

*From complex function theory to non-archimedean spaces - a number theoretical thread*

(Université Paris-Diderot)

**Thursday, ****June 3****, 202****1** *(**11am PDT, 2pm EDT, 7pm BST, 8pm CEST, 9pm Israel Daylight Time, 11:30pm Indian Standard Time**)*

**Friday, ****June 4****, 202****1** *(**2am CST, 4am AEST, 6am NZST**)*

**Thursday,**

**June 3**

**, 202**

**1**

*(*

*11am PDT, 2pm EDT, 7pm BST, 8pm CEST, 9pm Israel Daylight Time, 11:30pm Indian Standard Time*

*)*

**Friday,**

**June 4**

**, 202**

**1**

*(*

*2am CST, 4am AEST, 6am NZST*

*)*

*Abstract:* Diophantine geometry and complex function theory have a long and well known history of mutual friendship, attested, for example, by the fruitful interactions between height functions and potential theory. In the last 50 years, interactions even deepened with the invention of Arakelov geometry (Arakelov, Gillet/Soulé, Faltings) and its application by Szpiro/Ullmo/Zhang to equidistribution theorems and the Bogomolov conjecture. Roughly at the same time, Berkovich invented a new kind of non-archimedean analytic spaces which possess a rich and well behaved geometric structure. This opened the way to non-archimedean potential theory (Baker/Rumely, Favre/Rivera-Letelier), or to arithmetic/geometric equidistribution theorems in this case. More recently, Ducros and myself introduced basic ideas from tropical geometry and a construction of Lagerberg to construct a calculus of (p,q)-forms on Berkovich spaces, which is an analogue of the corresponding calculus on complex manifolds, and seems to be an attractive candidate for being the p-adic side of height function theory.

*Abstract:*Diophantine geometry and complex function theory have a long and well known history of mutual friendship, attested, for example, by the fruitful interactions between height functions and potential theory. In the last 50 years, interactions even deepened with the invention of Arakelov geometry (Arakelov, Gillet/Soulé, Faltings) and its application by Szpiro/Ullmo/Zhang to equidistribution theorems and the Bogomolov conjecture. Roughly at the same time, Berkovich invented a new kind of non-archimedean analytic spaces which possess a rich and well behaved geometric structure. This opened the way to non-archimedean potential theory (Baker/Rumely, Favre/Rivera-Letelier), or to arithmetic/geometric equidistribution theorems in this case. More recently, Ducros and myself introduced basic ideas from tropical geometry and a construction of Lagerberg to construct a calculus of (p,q)-forms on Berkovich spaces, which is an analogue of the corresponding calculus on complex manifolds, and seems to be an attractive candidate for being the p-adic side of height function theory.

### Link to recording (133MB)

Link to recording (133MB)

### Link to slides

Link to slides

## Robert Tichy, *Equidistribution, exponential sums and van der Corput sets *

(TU Graz)

Robert Tichy,

*Equidistribution, exponential sums and van der Corput sets*

(TU Graz)

**Thursday, ****May 27****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****May 28****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**May 27**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**May 28**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* The talk starts with a survey on Sarkoezy`s results on difference sets and with Furstenberg`s dynamic approach to additive problems. We present some results of a joint work with Bergelson, Kolesnik, Son and Madritsch concerning multidimensional van der Corput sets based on new bounds for exponential sums. In a second part we give a brief introduction on equidistribution theory focusing on the interplay of exponential sums with difference theorems. In a third part Hardy fields are discussed in some detail. This concept was introduced to equidistribution theory by Boshernitzan and it tuned out to be very fruitful. We will report on recent results of Bergelson et al. and at the very end on applications to diophantine approximation. This includes results concerning the approximation of polynomial-like functions along primes which were established in a joint work with Madritsch and sharpened very recently by my PhD student Minelli.

*Abstract:*The talk starts with a survey on Sarkoezy`s results on difference sets and with Furstenberg`s dynamic approach to additive problems. We present some results of a joint work with Bergelson, Kolesnik, Son and Madritsch concerning multidimensional van der Corput sets based on new bounds for exponential sums. In a second part we give a brief introduction on equidistribution theory focusing on the interplay of exponential sums with difference theorems. In a third part Hardy fields are discussed in some detail. This concept was introduced to equidistribution theory by Boshernitzan and it tuned out to be very fruitful. We will report on recent results of Bergelson et al. and at the very end on applications to diophantine approximation. This includes results concerning the approximation of polynomial-like functions along primes which were established in a joint work with Madritsch and sharpened very recently by my PhD student Minelli.

### Link to slides

Link to slides

## Alice Silverberg, *Cryptographic Multilinear Maps and Miscellaneous Musings*

(University of California, Irvine)

Alice Silverberg,

*Cryptographic Multilinear Maps and Miscellaneous Musings*

(University of California, Irvine)

**Thursday, ****May 20****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****May 21****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**May 20**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**May 21**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Recognizing that many of us have Zoom fatigue, I will keep this talk light, without too many technical details. In addition to discussing an open problem on multilinear maps that has applications to cryptography, I'll give miscellaneous musings about things I've learned over the years that I wish I'd learned sooner.

*Abstract:*Recognizing that many of us have Zoom fatigue, I will keep this talk light, without too many technical details. In addition to discussing an open problem on multilinear maps that has applications to cryptography, I'll give miscellaneous musings about things I've learned over the years that I wish I'd learned sooner.

### Link to recording (207MB)

Link to recording (207MB)

### Link to slides

Link to slides

## Alex Kontorovich, *Arithmetic Groups and Sphere Packings*

(Rutgers University)

Alex Kontorovich,

*Arithmetic Groups and Sphere Packings*

(Rutgers University)

**Thursday, ****May 13****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****May 14****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**May 13**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**May 14**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

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

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

### Link to recording (207MB)

Link to recording (207MB)

### Link to slides

Link to slides

## Akshay Venkatesh, *A brief history of Hecke operators*

(Institute for Advanced Study)

Akshay Venkatesh,

*A brief history of Hecke operators*

(Institute for Advanced Study)

**Thursday, ****May 6****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****May 7****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**May 6**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**May 7**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* This is an expository lecture about Hecke operators, in the context of number theory. We will trace some of the history of the ideas, starting before Hecke's birth and proceeding through the subsequent century. In particular we will discuss some of the original motivations and then the impact of ideas from representation theory and algebraic geometry. This lecture is aimed at non-experts.

*Abstract:*This is an expository lecture about Hecke operators, in the context of number theory. We will trace some of the history of the ideas, starting before Hecke's birth and proceeding through the subsequent century. In particular we will discuss some of the original motivations and then the impact of ideas from representation theory and algebraic geometry. This lecture is aimed at non-experts.

## Pietro Corvaja, *On the local-to-global principle for value sets*

(University of Udine)

Pietro Corvaja,

*On the local-to-global principle for value sets*

(University of Udine)

**Special Chair: Andrew Granville (University of Montreal)**

**Special Chair: Andrew Granville (University of Montreal)**

**Thursday, ****April 29****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****April 30****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**April 29**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**April 30**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Given a finite morphism f: X -> Y between algebraic curves over number fields, we study the set of rational (or integral) points in Y having a pre-image in every p-adic completion of the number field, but no rational pre-images. In particular, we investigate whether this set can be infinite.

*Abstract:*Given a finite morphism f: X -> Y between algebraic curves over number fields, we study the set of rational (or integral) points in Y having a pre-image in every p-adic completion of the number field, but no rational pre-images. In particular, we investigate whether this set can be infinite.

**We will mark the 1 year anniversary of the Number Theory Web Seminar. **

### Link to recording (102MB)

Link to recording (102MB)

## Renate Scheidler, *Computing modular polynomials and isogeny graphs of rank 2 Drinfeld modules*

(University of Calgary)

Renate Scheidler,

*Computing modular polynomials and isogeny graphs of rank 2 Drinfeld modules*

(University of Calgary)

**Thursday, ****April 22****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****April 23****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**April 22**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**April 23**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* Drinfeld modules represent the function field analogue of the theory of complex multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld in the 1970s in the course of proving the Langlands conjectures for GL(2) over global function fields. Drinfeld modules of rank 2 exhibit very similar behaviour to elliptic curves: they are classified as ordinary or supersingular, support isogenies and their duals, and their endomorphism rings have an analogous structure. Their isomorphism classes are parameterized by j-invariants, and Drinfeld modular polynomials can be used to compute their isogeny graphs whose ordinary connected components take the shape of volcanos. While the rich analytic and algebraic theory of Drinfeld modules has undergone extensive investigation, very little has been explored from a computational perspective. This research represents the first foray in this direction, introducing an algorithm for computing Drinfeld modular polynomials and isogeny graphs.

*Abstract:*Drinfeld modules represent the function field analogue of the theory of complex multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld in the 1970s in the course of proving the Langlands conjectures for GL(2) over global function fields. Drinfeld modules of rank 2 exhibit very similar behaviour to elliptic curves: they are classified as ordinary or supersingular, support isogenies and their duals, and their endomorphism rings have an analogous structure. Their isomorphism classes are parameterized by j-invariants, and Drinfeld modular polynomials can be used to compute their isogeny graphs whose ordinary connected components take the shape of volcanos. While the rich analytic and algebraic theory of Drinfeld modules has undergone extensive investigation, very little has been explored from a computational perspective. This research represents the first foray in this direction, introducing an algorithm for computing Drinfeld modular polynomials and isogeny graphs.

### This is joint work with Perlas Caranay and Matt Greenberg, as well as ongoing research with Edgar Pacheco Castan. Some familiarity with elliptic curves is expected for this talk, but no prior knowledge of Drinfeld modules is assumed.

This is joint work with Perlas Caranay and Matt Greenberg, as well as ongoing research with Edgar Pacheco Castan. Some familiarity with elliptic curves is expected for this talk, but no prior knowledge of Drinfeld modules is assumed.

### Link to recording (204MB)

Link to recording (204MB)

### Link to slides

Link to slides

## Jonathan Keating, *Joint Moments *

(University of Oxford)

Jonathan Keating,

*Joint Moments*

(University of Oxford)

**Thursday, ****April 15****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****April 16****, 202****1** *(**1am AEST, 3am NZST**)*

**Thursday,**

**April 15**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**April 16**

**, 202**

**1**

*(*

*1am AEST, 3am NZST*

*)*

*Abstract:* I will discuss the joint moments of the Riemann zeta-function and its derivative, and the corresponding joint moments of the characteristic polynomials of random unitary matrices and their derivatives.

*Abstract:*I will discuss the joint moments of the Riemann zeta-function and its derivative, and the corresponding joint moments of the characteristic polynomials of random unitary matrices and their derivatives.

### Link to recording (105MB)

Link to recording (105MB)

### Link to slides

Link to slides

## János Pintz, *On the mean value of the remainder term of the prime number formula *

(Alfréd Rényi Institute of Mathematics)

János Pintz,

*On the mean value of the remainder term of the prime number formula*

(Alfréd Rényi Institute of Mathematics)

**Thursday, ****April 8****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****April 9****, 202****1** *(**1am AEST, 3am NZDT**)*

**Thursday,**

**April 8**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**April 9**

**, 202**

**1**

*(*

*1am AEST, 3am NZDT*

*)*

*Abstract:* There are several methods to obtain a lower bound for the mean value of the absolute value of the remainder term of the prime number formula as function of a hypothetical zero of the Riemann Zeta function off the critical line. (The case when the Riemann Hypothesis is true can be treated easier.) The most efficient ones include results of Knapowski-Turán, Sz. Gy. Révész , and the author, proved by several different methods.

*Abstract:*There are several methods to obtain a lower bound for the mean value of the absolute value of the remainder term of the prime number formula as function of a hypothetical zero of the Riemann Zeta function off the critical line. (The case when the Riemann Hypothesis is true can be treated easier.) The most efficient ones include results of Knapowski-Turán, Sz. Gy. Révész , and the author, proved by several different methods.

### The result to be proved in the lecture provides (again with an other method) a quite good lower bound and it has the good feature (which is useful in further applications too) that instead of the whole interval [0,X] it gives a good lower bound for the average on [F(X), X] with log F(X) close to log X (that is on "short" intervals measured with the logarithmic scale).

The result to be proved in the lecture provides (again with an other method) a quite good lower bound and it has the good feature (which is useful in further applications too) that instead of the whole interval [0,X] it gives a good lower bound for the average on [F(X), X] with log F(X) close to log X (that is on "short" intervals measured with the logarithmic scale).

### Link to recording (111MB)

Link to recording (111MB)

### Link to slides

Link to slides

## Boris Adamczewski, *Furstenberg's conjecture, Mahler's method, and finite automata*

(CNRS, Université Claude Bernard Lyon 1)

Boris Adamczewski,

*Furstenberg's conjecture, Mahler's method, and finite automata*

(CNRS, Université Claude Bernard Lyon 1)

**Thursday, ****April 1****, 202****1** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm **Israel Daylight Time**, 8:30pm Indian Standard Time, 11pm CST)*

**Friday, ****April 2****, 202****1** *(**2am AEDT, 4am NZDT**)*

**Thursday,**

**April 1**

**, 202**

**1**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm*

*Israel Daylight Time*

*, 8:30pm Indian Standard Time, 11pm CST)*

**Friday,**

**April 2**

**, 202**

**1**

*(*

*2am AEDT, 4am NZDT*

*)*

*Abstract:* It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.

*Abstract:*It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.

### Link to recording (96MB)

Link to recording (96MB)

### Link to slides

Link to slides

## Vitaly Bergelson, *A "soft" dynamical approach to the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions*

(Ohio State University)

Vitaly Bergelson,

*A "soft" dynamical approach to the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions*

(Ohio State University)

**Thursday, March 25, 2021 **(*1:30pm PDT, 4:30pm EDT, 8:30pm GMT, 9:30pm CET, 10:30pm Israel Standard Time*)

**Thursday, March 25, 2021**(

*1:30pm PDT, 4:30pm EDT, 8:30pm GMT, 9:30pm CET, 10:30pm Israel Standard Time*)

**Friday, March 26, 2021 **(*2am Indian Standard Time, 4:30am CST, 7:30am AEDT, 9:30am NZDT*)

**Friday, March 26, 2021**(

*2am Indian Standard Time, 4:30am CST, 7:30am AEDT, 9:30am NZDT*)

*Abstract:* We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdős-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,×).

*Abstract:*We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdős-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak’s Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,×).

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

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

### Link to recording (191MB)

Link to recording (191MB)

### Link to slides

Link to slides

## Shabnam Akhtari, *Orders in Quartic Number Fields and Classical Diophantine Equations*

(University of Oregon)

Shabnam Akhtari,

*Orders in Quartic Number Fields and Classical Diophantine Equations*

(University of Oregon)

**Thursday, March 18, 2021 ***(**1pm P**D**T, **4**pm E**D**T, 8pm GMT, 9pm CET, 10pm Israel Standard Time**)*

**Thursday, March 18, 2021**

*(*

*1pm P*

*D*

*T,*

*4*

*pm E*

*D*

*T, 8pm GMT, 9pm CET, 10pm Israel Standard Time*

*)*

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

**Friday, March 19, 2021**

*(*

*1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT*

*)*

### Link to recording (121MB)

Link to recording (121MB)

### Link to slides

Link to slides

## Chantal David, *Moments and non-vanishing of cubic Dirichlet L-functions at s=1/2*

(Concordia University)

Chantal David,

*Moments and non-vanishing of cubic Dirichlet L-functions at s=1/2*

(Concordia University)

**Thursday, March 11, 2021 ***(**12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time**)*

**Thursday, March 11, 2021**

*(*

*12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time*

*)*

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

**Friday, March 12, 2021**

*(*

*1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT*

*)*

### Link to recording (110MB)

Link to recording (110MB)

## Sanju Velani, *The Shrinking Target Problem for Matrix Transformations of Tori*

(University of York)

Sanju Velani,

*The Shrinking Target Problem for Matrix Transformations of Tori*

(University of York)

**Thursday, March 4, 2021 ***(**12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time**)*

**Thursday, March 4, 2021**

*(*

*12pm PST, 3pm EST, 8pm GMT, 9pm CET, 10pm Israel Standard Time*

*)*

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

**Friday, March 5, 2021**

*(*

*1:30am Indian Standard Time, 4am CST, 7am AEDT, 9am NZDT*

*)*

### Link to recording (112MB)

Link to recording (112MB)

### Link to slides

Link to slides

## Ben Green, *New lower bounds for van der Waerden numbers*

(University of Oxford)

Ben Green,

*New lower bounds for van der Waerden numbers*

(University of Oxford)

**Thursday, February 25, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, February 25, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, February 26, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, February 26, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* Colour {1,..,N} red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about sqrt{N}. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about exp ((log N)^{3/4}), and in particular less than any fixed power of N.

*Abstract:*Colour {1,..,N} red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about sqrt{N}. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about exp ((log N)^{3/4}), and in particular less than any fixed power of N.

### I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory.

I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory.

### Link to recording (143MB)

Link to recording (143MB)

### Link to slides

Link to slides

## Gabriel Dill, *Unlikely Intersections and Distinguished Categories*

(University of Oxford)

Gabriel Dill,

*Unlikely Intersections and Distinguished Categories*

(University of Oxford)

**Thursday, February 18, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, February 18, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, February 19, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, February 19, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* After a general introduction to the field of unlikely intersections, I present current work in progress with Fabrizio Barroero, in which we propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows to us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This also yields new unconditional results on the Zilber-Pink conjecture for curves in various contexts.

*Abstract:*After a general introduction to the field of unlikely intersections, I present current work in progress with Fabrizio Barroero, in which we propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows to us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This also yields new unconditional results on the Zilber-Pink conjecture for curves in various contexts.

### Link to recording (78MB)

Link to recording (78MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Don Zagier, *Analytic functions related to zeta-values, cotangent products, and the cohomology of SL_2(\Z)*

(Max Planck Institute for Mathematics)

Don Zagier,

*Analytic functions related to zeta-values, cotangent products, and the cohomology of SL_2(\Z)*

(Max Planck Institute for Mathematics)

**Thursday, February 11, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, February 11, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, February 12, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, February 12, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* I will report on the properties of various functions, going back essentially to Herglotz, that relate to a number of different topics in number theory, including those in the title but also others like Hecke operators or Stark's conjectures. This is joint work with Danylo Radchenko.

*Abstract:*I will report on the properties of various functions, going back essentially to Herglotz, that relate to a number of different topics in number theory, including those in the title but also others like Hecke operators or Stark's conjectures. This is joint work with Danylo Radchenko.

### Link to recording (326MB)

Link to recording (326MB)

## Oleksiy Klurman, *On the zeros of Fekete polynomials*

(University of Bristol)

Oleksiy Klurman,

*On the zeros of Fekete polynomials*

(University of Bristol)

**Thursday, February 4, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, February 4, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, February 5, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, February 5, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem. The goal of this talk is to discuss what we knew, know and would like to know about zeros of such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.

*Abstract:*Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem. The goal of this talk is to discuss what we knew, know and would like to know about zeros of such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.

### Link to recording (81MB)

Link to recording (81MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## William Banks, *On the distribution of reduced fractions with squarefree denominators*

(University of Missouri)

William Banks,

*On the distribution of reduced fractions with squarefree denominators*

(University of Missouri)

**Thursday, January 28, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, January 28, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, January 29, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, January 29, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* In this talk we discuss how the nonvanishing of the Riemann zeta function in a half-plane {sigma>sigma_0}, with some real sigma_0<1, is equivalent to a strong statement about the distribution in the unit interval of reduced fractions with squarefree denominators.

*Abstract:*In this talk we discuss how the nonvanishing of the Riemann zeta function in a half-plane {sigma>sigma_0}, with some real sigma_0<1, is equivalent to a strong statement about the distribution in the unit interval of reduced fractions with squarefree denominators.

### The approach utilizes an unconditional generalization of a theorem of Blomer concerning the distribution "on average" of squarefree integers in arithmetic progressions to large moduli.

The approach utilizes an unconditional generalization of a theorem of Blomer concerning the distribution "on average" of squarefree integers in arithmetic progressions to large moduli.

### Link to recording (143MB)

Link to recording (143MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Lior Bary-Soroker, *Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic*

(Tel Aviv University)

Lior Bary-Soroker,

*Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic*

(Tel Aviv University)

**Thursday, January 21, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, January 21, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, January 22, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, January 22, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* In the talk we will discuss recent advances on the following two questions:

*Abstract:*In the talk we will discuss recent advances on the following two questions:

### Let A(X) = sum ±X^i be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.

Let A(X) = sum ±X^i be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.

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

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

### Q2: What is the typical Galois group of A?

Q2: What is the typical Galois group of A?

### One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.

One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.

### Link to recording (333MB)

Link to recording (333MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Peter Sarnak, *Summation formulae in spectral theory and number theory *

(Institute for Advanced Study and Princeton University)

Peter Sarnak,

*Summation formulae in spectral theory and number theory*

(Institute for Advanced Study and Princeton University)

**A talk in honor of Zeev Rudnick's 60th birthday**

**Special Chair: Lior Bary-Soroker**

**Special Chair: Lior Bary-Soroker**

**Thursday, January 14, 2021 **(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Thursday, January 14, 2021**(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)

**Friday, January 15, 2021 **(12am CST, 3am AEDT, 5am NZDT)

**Friday, January 15, 2021**(12am CST, 3am AEDT, 5am NZDT)

*Abstract:* The Poisson Summation formula, Riemann-Guinand-Weil explicit formula, Selberg Trace Formula and Lefschetz Trace formula in the function field, are starting points for a number of Zeev Rudnick's works. We will review some of these before describing some recent applications (joint with P. Kurasov) of Lang's G_m conjectures to the additive structure of the spectra of metric graphs and crystalline measures.

*Abstract:*The Poisson Summation formula, Riemann-Guinand-Weil explicit formula, Selberg Trace Formula and Lefschetz Trace formula in the function field, are starting points for a number of Zeev Rudnick's works. We will review some of these before describing some recent applications (joint with P. Kurasov) of Lang's G_m conjectures to the additive structure of the spectra of metric graphs and crystalline measures.

### Link to recording (117MB)

Link to recording (117MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Imre Ruzsa, *Additive decomposition of signed primes*

(Alfréd Rényi Institute of Mathematics)

Imre Ruzsa,

*Additive decomposition of signed primes*

(Alfréd Rényi Institute of Mathematics)

**Thursday, ****January ****7, 202****1** *(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm **I**srael Standard Time**, 9:30pm IST)*

**Friday, ****January ****8, 202****1** *(12am CST, 3am AEDT, 5am NZDT)*

**Thursday,**

**January**

**7, 202**

**1**

*(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm*

*I*

*srael Standard Time*

*, 9:30pm IST)*

**Friday,**

**January**

**8, 202**

**1**

*(12am CST, 3am AEDT, 5am NZDT)*

*Abstract:* Assuming the prime-tuple hypothesis, the set of signed primes is a sumset. More exactly, there are infinite sets A, B of integers such that A+B consists exactly of the (positive and negative) primes with |p|>3. I will also meditate on the possibility of a triple sum and analogous problems for the set of squarefree numbers.

*Abstract:*Assuming the prime-tuple hypothesis, the set of signed primes is a sumset. More exactly, there are infinite sets A, B of integers such that A+B consists exactly of the (positive and negative) primes with |p|>3. I will also meditate on the possibility of a triple sum and analogous problems for the set of squarefree numbers.

### Link to recording (63MB)

Link to recording (63MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Jianya Liu, *Mobius disjointness for irregular flows *

(Shandong University)

Jianya Liu,

*Mobius disjointness for irregular flows*

(Shandong University)

**Tuesday, December 22, 2020** *(**2am PST, 5am EST, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, December 22, 2020**

*(*

*2am PST, 5am EST, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

*Abstract:* The behavior of the Mobius function is central in the theory of prime numbers. A surprising connection with the theory of dynamical systems was discovered in 2010 by P. Sarnak, who formulated the Mobius Disjointness Conjecture (MDC), which asserts that the Mobius function is linearly disjoint from any zero-entropy flows. This conjecture opened the way into a large body of research on the interface of analytic number theory and ergodic theory. In this talk I will report how to establish MDC for a class of irregular flows, which are in general mysterious and ill understood. This is based on joint works with P. Sarnak, and with W. Huang and K. Wang.

*Abstract:*The behavior of the Mobius function is central in the theory of prime numbers. A surprising connection with the theory of dynamical systems was discovered in 2010 by P. Sarnak, who formulated the Mobius Disjointness Conjecture (MDC), which asserts that the Mobius function is linearly disjoint from any zero-entropy flows. This conjecture opened the way into a large body of research on the interface of analytic number theory and ergodic theory. In this talk I will report how to establish MDC for a class of irregular flows, which are in general mysterious and ill understood. This is based on joint works with P. Sarnak, and with W. Huang and K. Wang.

### Link to recording (71MB)

Link to recording (71MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Gisbert Wüstholz, *Baker's theory for 1-motives*

(ETH / University Zurich)

Gisbert Wüstholz,

*Baker's theory for 1-motives*

(ETH / University Zurich)

**Thursday, December 17, 2020** *(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm **I**srael Standard Time**, 9:30pm IST)*

**Friday, December 18, 2020** *(12am CST, 3am AEDT, 5am NZDT)*

**Thursday, December 17, 2020**

*(8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm*

*I*

*srael Standard Time*

*, 9:30pm IST)*

**Friday, December 18, 2020**

*(12am CST, 3am AEDT, 5am NZDT)*

*Abstract:* From a historical point of view transcendence theory used to be a nice collection of mostly particular results, very difficult to find and to prove. To find numbers for which one has a chance to prove transcendence is very difficult. To state conjecture is not so difficult but in most cases hopeless to prove. In our lecture we try to draw a picture of quite far reaching frames in the theory of motives which can put transcendence theory into a more conceptual setting.

*Abstract:*From a historical point of view transcendence theory used to be a nice collection of mostly particular results, very difficult to find and to prove. To find numbers for which one has a chance to prove transcendence is very difficult. To state conjecture is not so difficult but in most cases hopeless to prove. In our lecture we try to draw a picture of quite far reaching frames in the theory of motives which can put transcendence theory into a more conceptual setting.

### Looking at periods of rational 1-forms on varieties we realized that there is a more conceptual background behind the properties of these complex numbers than had been thought so far. The central question which I was trying for more than three decades to answer was to determine when a period is algebraic. A priori a period is zero, algebraic or transcendental, no surprise! It is also not difficult to give examples for cases when periods are algebraic. However the big question was whether the examples are all examples. Quite recently, partly jointly with Annette Huber we developed a new transcendence theory within 1-motives which extend commutative algebraic groups. One outcome was that algebraicity of periods has a very conceptual description and we shall give a precise and surprisingly simple answer.

Looking at periods of rational 1-forms on varieties we realized that there is a more conceptual background behind the properties of these complex numbers than had been thought so far. The central question which I was trying for more than three decades to answer was to determine when a period is algebraic. A priori a period is zero, algebraic or transcendental, no surprise! It is also not difficult to give examples for cases when periods are algebraic. However the big question was whether the examples are all examples. Quite recently, partly jointly with Annette Huber we developed a new transcendence theory within 1-motives which extend commutative algebraic groups. One outcome was that algebraicity of periods has a very conceptual description and we shall give a precise and surprisingly simple answer.

### Many questions which were central in transcendence theory and with a long and famous history turn out to get a general answer within the new theory. The classical work of Baker turns out to be a very special but seminal case.

Many questions which were central in transcendence theory and with a long and famous history turn out to get a general answer within the new theory. The classical work of Baker turns out to be a very special but seminal case.

### Link to recording (161MB)

Link to recording (161MB)

## Adam Harper, *Large fluctuations of random multiplicative functions*

(University of Warwick)

Adam Harper,

*Large fluctuations of random multiplicative functions*

(University of Warwick)

**Tuesday, December 15, 2020** *(**2am PST, 5am EST, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, December 15, 2020**

*(*

*2am PST, 5am EST, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

*Abstract:* Random multiplicative functions $f(n)$ are a well studied random model for deterministic multiplicative functions like Dirichlet characters or the Mobius function. Arguably the first question ever studied about them, by Wintner in 1944, was to obtain almost sure bounds for the largest fluctuations of their partial $\sum_{n \leq x} f(n)$, seeking to emulate the classical Law of the Iterated Logarithm for independent random variables. It remains an open question to sharply determine the size of these fluctuations, and in this talk I will describe a new result in that direction. I hope to get to some interesting details of the new proof in the latter part of the talk, but most of the discussion should be widely accessible.

*Abstract:*Random multiplicative functions $f(n)$ are a well studied random model for deterministic multiplicative functions like Dirichlet characters or the Mobius function. Arguably the first question ever studied about them, by Wintner in 1944, was to obtain almost sure bounds for the largest fluctuations of their partial $\sum_{n \leq x} f(n)$, seeking to emulate the classical Law of the Iterated Logarithm for independent random variables. It remains an open question to sharply determine the size of these fluctuations, and in this talk I will describe a new result in that direction. I hope to get to some interesting details of the new proof in the latter part of the talk, but most of the discussion should be widely accessible.

### Link to recording (91MB)

Link to recording (91MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Maksym Radziwill, *The Fyodorov-Hiary-Keating conjecture*

(California Institute of Technology)

Maksym Radziwill,

*The Fyodorov-Hiary-Keating conjecture*

(California Institute of Technology)

**Thursday, December 10, 2020** *(**8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I**srael Standard Time**, 9:30pm IST**)*

**Friday, December 11, 2020*** (12am China Standard Time, 3am AEDT, 5am NZDT)*

**Thursday, December 10, 2020**

*(*

*8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I*

*srael Standard Time*

*, 9:30pm IST*

*)*

**Friday, December 11, 2020**

*(12am China Standard Time, 3am AEDT, 5am NZDT)*

*Abstract:* I will discuss recent progress on the Fyodorov-Hiary-Keating conjecture on the distribution of the local maximum of the Riemann zeta-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.

*Abstract:*I will discuss recent progress on the Fyodorov-Hiary-Keating conjecture on the distribution of the local maximum of the Riemann zeta-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.

### Link to recording (76MB)

Link to recording (76MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

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

Jacob Tsimerman, Bounding torsion in class group and families of local systems

*(University of Toronto)*

**Please note the unusual time!**

**Please note the unusual time!**

**Monday, December 7, 2020** (2pm PST, 5pm EST, 10pm GMT, 11pm CET)

**Tuesday, December 8, 2020** (12am Israel Standard Time, 3:30am Indian Standard Time, 6am CST, 9am AEDT, 11am NZDT)

**Monday, December 7, 2020**(2pm PST, 5pm EST, 10pm GMT, 11pm CET)

**Tuesday, December 8, 2020**(12am Israel Standard Time, 3:30am Indian Standard Time, 6am CST, 9am AEDT, 11am NZDT)

*Abstract:* (joint w/ Arul Shankar) We discuss a new method to bound 5-torsion in class groups of quadratic fields using the refined BSD conjecture for elliptic curves. The most natural “trivial” bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a selmer group of a Galois module. We may then similarly bound its size by the Tate-Shafarevich group of an appropriate elliptic curve, which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds selmer groups of finite Galois modules by embedding into global objects, and using class number formulas. If time permits, we explain how the function field picture yields unconditional results and suggests further generalizations.

*Abstract:*(joint w/ Arul Shankar) We discuss a new method to bound 5-torsion in class groups of quadratic fields using the refined BSD conjecture for elliptic curves. The most natural “trivial” bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a selmer group of a Galois module. We may then similarly bound its size by the Tate-Shafarevich group of an appropriate elliptic curve, which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds selmer groups of finite Galois modules by embedding into global objects, and using class number formulas. If time permits, we explain how the function field picture yields unconditional results and suggests further generalizations.

### Link to recording (86MB)

Link to recording (86MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Alexander Lubotzky, *From Ramanujan graphs to Ramanujan complexes** *

(Hebrew University of Jerusalem)

Alexander Lubotzky,

*From Ramanujan graphs to Ramanujan complexes*

*(Hebrew University of Jerusalem)*

**Thursday, December 3, 2020** *(**8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I**srael Standard Time**, 9:30pm IST**)*

**Friday, December 4, 2020*** (12am China Standard Time, 3am AEDT, 5am NZDT)*

**Thursday, December 3, 2020**

*(*

*8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I*

*srael Standard Time*

*, 9:30pm IST*

*)*

**Friday, December 4, 2020**

*(12am China Standard Time, 3am AEDT, 5am NZDT)*

*Abstract:* Ramanujan graphs are k-regular graphs with all non trivial eigenvalues bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups. The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms.

*Abstract:*Ramanujan graphs are k-regular graphs with all non trivial eigenvalues bounded (in absolute value) by 2SR(k-1). They are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by the action of suitable congruence subgroups of arithmetic groups. The spectral bound was proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms.

### The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplicial complexes which on one hand are "random like" and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.

The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings associated with GL(n) over F. This way one gets finite simplicial complexes which on one hand are "random like" and at the same time have strong symmetries. These seemingly contradicting properties make them very useful for constructions of various external objects.

### Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will survey some of these applications.

Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will survey some of these applications.

### Link to recording (174MB)

Link to recording (174MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Dragos Ghioca, *A couple of conjectures in arithmetic dynamics over fields of** **positive characteristic*

(University of British Columbia)

Dragos Ghioca,

*A couple of conjectures in arithmetic dynamics over fields of*

*positive characteristic*

(University of British Columbia)

**Monday, November 30, 2020 ****(***5pm PST, 8pm EST**)*

**Tuesday, December 1, 2020 ****(***1am GMT, 2am CET, 3am I**srael Standard Time**, 6:30am **IST**, 9am C**hina Standard Time**, 12pm AEDT, 2pm NZDT**)*

**Monday, November 30, 2020**

**(***5pm PST, 8pm EST*

*)*

**Tuesday, December 1, 2020**

**(***1am GMT, 2am CET, 3am I*

*srael Standard Time*

*, 6:30am*

*IST*

*, 9am C*

*hina Standard Time*

*, 12pm AEDT, 2pm NZDT*

*)*

*Abstract:* The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.

*Abstract:*The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.

### Link to recording (75MB)

Link to recording (75MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Michael Stoll, *An application of "Selmer group Chabauty" to arithmetic dynamics *

(University of Bayreuth)

Michael Stoll,

*An application of "Selmer group Chabauty" to arithmetic dynamics*

(University of Bayreuth)

**Thursday, November 26, 2020** *(**8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I**srael Standard Time**, 9:30pm IST**)*

**Friday, ****November 27****, 2020*** (12am China Standard Time, **3am AEDT, 5am NZDT**)*

**Thursday, November 26, 2020**

*(*

*8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I*

*srael Standard Time*

*, 9:30pm IST*

*)*

**Friday,**

**November 27**

**, 2020**

*(12am China Standard Time,*

*3am AEDT, 5am NZDT*

*)*

*Abstract:* The irreducibility or otherwise of iterates of polynomials is an important question in arithmetic dynamics. For example, it is conjectured that whenever the second iterate of x^2 + c (with c a rational number) is irreducible over Q, then so are all iterates. A sufficient criterion for the iterates to be irreducible can be expressed in terms of rational points on certain hyperelliptic curves. We will show how to use the "Selmer group Chabauty" method developed by the speaker to determine the set of rational points on a hyperelliptic curve of genus 7. This leads to a proof that the seventh iterate of x^2 + c must be irreducible if the second iterate is. Assuming GRH, we can extend this to the tenth iterate.

*Abstract:*The irreducibility or otherwise of iterates of polynomials is an important question in arithmetic dynamics. For example, it is conjectured that whenever the second iterate of x^2 + c (with c a rational number) is irreducible over Q, then so are all iterates. A sufficient criterion for the iterates to be irreducible can be expressed in terms of rational points on certain hyperelliptic curves. We will show how to use the "Selmer group Chabauty" method developed by the speaker to determine the set of rational points on a hyperelliptic curve of genus 7. This leads to a proof that the seventh iterate of x^2 + c must be irreducible if the second iterate is. Assuming GRH, we can extend this to the tenth iterate.

### Link to recording (110MB)

Link to recording (110MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Jasmin Matz, *Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect*

(University of Copenhagen)

Jasmin Matz,

*Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect*

(University of Copenhagen)

**Tuesday, November 24, 2020** *(**2am PST, 5am EST, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, November 24, 2020**

*(*

*2am PST, 5am EST, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

*Abstract:* Suppose M is a closed Riemannian manifold with an orthonormal basis B of L^2(M) consisting of Laplace eigenfunctions. A classical result of Shnirelman and others proves that if the geodesic flow on the cotangent bundle of M is ergodic, then M is quantum ergodic, in particular, on average, the probability measures defined by the functions f in B on M tends on average towards the Riemannian measure on M in the high energy limit (i.e, as the Laplace eigenvalues of f -> infinity).

We now want to look at a level aspect of this property, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds M_j together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of SL(n,R)/SO(n).

*Abstract:*Suppose M is a closed Riemannian manifold with an orthonormal basis B of L^2(M) consisting of Laplace eigenfunctions. A classical result of Shnirelman and others proves that if the geodesic flow on the cotangent bundle of M is ergodic, then M is quantum ergodic, in particular, on average, the probability measures defined by the functions f in B on M tends on average towards the Riemannian measure on M in the high energy limit (i.e, as the Laplace eigenvalues of f -> infinity).

We now want to look at a level aspect of this property, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds M_j together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of SL(n,R)/SO(n).

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Jason Bell, *A transcendental dynamical degree*

(University of Waterloo)

Jason Bell,

*A transcendental dynamical degree*

(University of Waterloo)

**Monday, November 16, 2020 ****(***5pm PST, 8pm EST**)*

**Tuesday, November 17, 2020 ****(***1am GMT, 2am CET, 3am I**srael Standard Time**, 6:30am I**ST**, 9am C**hina Standard Time**, 12pm AEDT, 2pm NZDT**)*

**Monday, November 16, 2020**

**(***5pm PST, 8pm EST*

*)*

**Tuesday, November 17, 2020**

**(***1am GMT, 2am CET, 3am I*

*srael Standard Time*

*, 6:30am I*

*ST*

*, 9am C*

*hina Standard Time*

*, 12pm AEDT, 2pm NZDT*

*)*

*Abstract:* The degree of a dominant rational map $f:\mathbb{P}^n\to \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\deg(f^n)$, which is submultiplicative and hence has the property that there is some $\lambda\ge 1$ such that $(\deg(f^n))^{1/n}\to \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We’ll give an overview of the significance of the dynamical degree in complex dynamics and describe an example in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller and Mattias Jonsson.

*Abstract:*The degree of a dominant rational map $f:\mathbb{P}^n\to \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\deg(f^n)$, which is submultiplicative and hence has the property that there is some $\lambda\ge 1$ such that $(\deg(f^n))^{1/n}\to \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We’ll give an overview of the significance of the dynamical degree in complex dynamics and describe an example in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller and Mattias Jonsson.

### Link to recording (96MB)

Link to recording (96MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## David Masser, *Pencils of norm form equations and a conjecture of Thomas*

(University of Basel)

David Masser,

*Pencils of norm form equations and a conjecture of Thomas*

(University of Basel)

**Thursday, November 12, 2020** *(**8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I**srael Standard Time**, 9:30pm IST**)*

**Friday, ****November 13,**** 2020*** (12am China Standard Time, 3am AEDT, 5am NZDT)*

**Thursday, November 12, 2020**

*(*

*8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I*

*srael Standard Time*

*, 9:30pm IST*

*)*

**Friday,**

**November 13,**

**2020**

*(12am China Standard Time, 3am AEDT, 5am NZDT)*

*Abstract:* We consider certain one-parameter families of norm form (and other) diophantine equations, and we solve them completely and uniformly for all sufficiently large positive integer values of the parameter (everything effective), following a line started by Emery Thomas in 1990. The new tool is a bounded height result from 2017 by Francesco Amoroso, Umberto Zannier and the speaker.

*Abstract:*We consider certain one-parameter families of norm form (and other) diophantine equations, and we solve them completely and uniformly for all sufficiently large positive integer values of the parameter (everything effective), following a line started by Emery Thomas in 1990. The new tool is a bounded height result from 2017 by Francesco Amoroso, Umberto Zannier and the speaker.

### Link to recording (124MB)

Link to recording (124MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Gérald Tenenbaum, *Recent progress on the Selberg-Delange method in analytic number theory*

(Université de Lorraine)

Gérald Tenenbaum,

*Recent progress on the Selberg-Delange method in analytic number theory*

(Université de Lorraine)

**Tuesday, November 10, 2020** *(**2am PST, 5am EST, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, November 10, 2020**

*(*

*2am PST, 5am EST, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

### Link to recording (94MB)

Link to recording (94MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Pär Kurlberg, *Distribution of lattice points on hyperbolic circles** *

(KTH)

Pär Kurlberg,

*Distribution of lattice points on hyperbolic circles*

*(KTH)*

**Thursday, November 5, 2020** *(**8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I**srael Standard Time**, 9:30pm IST**)*

**Friday, ****November 6****, 2020*** (12am China Standard Time, 3am AEDT, 5am NZDT)*

**Thursday, November 5, 2020**

*(*

*8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm I*

*srael Standard Time*

*, 9:30pm IST*

*)*

**Friday,**

**November 6**

**, 2020**

*(12am China Standard Time, 3am AEDT, 5am NZDT)*

*Abstract:* We study the distribution of lattice points lying on expanding circles in the hyperbolic plane. The angles of lattice points arising from the orbit of the modular group PSL(2,Z), and lying on hyperbolic circles centered at i, are shown to be equidistributed for generic radii (among the ones that contain points). We also show that angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of euclidean lattice points lying on circles in the plane, along a thin subsequence of radii. This is joint work with D. Chatzakos, S. Lester and I. Wigman.

*Abstract:*We study the distribution of lattice points lying on expanding circles in the hyperbolic plane. The angles of lattice points arising from the orbit of the modular group PSL(2,Z), and lying on hyperbolic circles centered at i, are shown to be equidistributed for generic radii (among the ones that contain points). We also show that angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of euclidean lattice points lying on circles in the plane, along a thin subsequence of radii. This is joint work with D. Chatzakos, S. Lester and I. Wigman.

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Jens Marklof, *The three gap theorem in higher dimensions*

(University of Bristol)

Jens Marklof,

*The three gap theorem in higher dimensions*

(University of Bristol)

**Tuesday, November 3, 2020** *(**2am PST, 5am EST, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, November 3, 2020**

*(*

*2am PST, 5am EST, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

*Abstract:* Take a point on the unit circle and rotate it N times by a fixed angle. The N points thus generated partition the circle into N intervals. A beautiful fact, first conjectured by Hugo Steinhaus in the 1950s and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski, is that for any choice of N, no matter how large, these intervals can have at most three distinct lengths. In this lecture I will explore an interpretation of the three gap theorem in terms of the space of Euclidean lattices, which will produce various new results in higher dimensions, including gaps in the fractional parts of linear forms and nearest neighbour distances in multi-dimensional Kronecker sequences. The lecture is based on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).

*Abstract:*Take a point on the unit circle and rotate it N times by a fixed angle. The N points thus generated partition the circle into N intervals. A beautiful fact, first conjectured by Hugo Steinhaus in the 1950s and proved independently by Vera Sós, János Surányi and Stanisław Świerczkowski, is that for any choice of N, no matter how large, these intervals can have at most three distinct lengths. In this lecture I will explore an interpretation of the three gap theorem in terms of the space of Euclidean lattices, which will produce various new results in higher dimensions, including gaps in the fractional parts of linear forms and nearest neighbour distances in multi-dimensional Kronecker sequences. The lecture is based on joint work with Alan Haynes (Houston) and Andreas Strömbergsson (Uppsala).

- J. Marklof and A. Strömbergsson, The three gap theorem and the space of lattices, American Mathematical Monthly 124 (2017) 741-745 https://people.maths.bris.ac.uk/~majm/bib/threegap.pdf
- A. Haynes and J. Marklof, Higher dimensional Steinhaus and Slater problems via homogeneous dynamics, Annales scientifiques de l'Ecole normale superieure 53 (2020) 537-557 https://people.maths.bris.ac.uk/~majm/bib/steinhaus.pdf
- A. Haynes and J. Marklof, A five distance theorem for Kronecker sequences, preprint arXiv:2009.08444 https://people.maths.bris.ac.uk/~majm/bib/steinhaus2.pdf

### Link to recording (104MB)

Link to recording (104MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Will Sawin, *The distribution of prime polynomials over finite fields*

(Columbia University)

Will Sawin,

*The distribution of prime polynomials over finite fields*

(Columbia University)

**Thursday, October 29, 2020** *(**9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm I**srael Standard **Time, 9:30pm IST**)*

**Friday, October 30, 2020*** (12am China Standard Time, 3am AEDT, 5am NZDT)*

**Thursday, October 29, 2020**

*(*

*9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm I*

*srael Standard*

*Time, 9:30pm IST*

*)*

**Friday, October 30, 2020**

*(12am China Standard Time, 3am AEDT, 5am NZDT)*

*Abstract:* Many conjectures in number theory have analogues for polynomials in one variable over a finite field. In recent works with Mark Shusterman, we proved analogues of two conjectures about prime numbers - the twin primes conjecture and the conjecture that there are infinitely many primes of the form n²+1. I will describe these results and explain some of the key ideas in the proofs, which combine classical analytic methods, elementary algebraic manipulations, and geometric methods.

*Abstract:*Many conjectures in number theory have analogues for polynomials in one variable over a finite field. In recent works with Mark Shusterman, we proved analogues of two conjectures about prime numbers - the twin primes conjecture and the conjecture that there are infinitely many primes of the form n²+1. I will describe these results and explain some of the key ideas in the proofs, which combine classical analytic methods, elementary algebraic manipulations, and geometric methods.

### Link to recording (109MB)

Link to recording (109MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Gal Binyamini, *Point counting for foliations in Diophantine geometry*

(Weizmann Institute of Science)

Gal Binyamini,

*Point counting for foliations in Diophantine geometry*

(Weizmann Institute of Science)

**Tuesday, October 27, 2020** *(**3am PDT, 6am EDT, 10am GMT, 11am CET, 12pm I**srael Standard Time**, 3:30pm IST, 6pm C**hina Standard Time**, 9pm AEDT, 11pm NZDT**)*

**Tuesday, October 27, 2020**

*(*

*3am PDT, 6am EDT, 10am GMT, 11am CET, 12pm I*

*srael Standard Time*

*, 3:30pm IST, 6pm C*

*hina Standard Time*

*, 9pm AEDT, 11pm NZDT*

*)*

*Abstract:* I will discuss "point counting" in two broad senses: counting the intersections between a trascendental variety and an algebraic one; and counting the number of algebraic points, as a function of degree and height, on a transcendental variety. After reviewing the fundamental results in this area - from the theory of o-minimal structures and the Pila-Wilkie theorem, I will restrict attention to the case that the transcendental variety is given in terms of a leaf of an algebraic foliation, and everything is defined over a number field. It turns out that in this case far stronger estimates can be obtained.

*Abstract:*I will discuss "point counting" in two broad senses: counting the intersections between a trascendental variety and an algebraic one; and counting the number of algebraic points, as a function of degree and height, on a transcendental variety. After reviewing the fundamental results in this area - from the theory of o-minimal structures and the Pila-Wilkie theorem, I will restrict attention to the case that the transcendental variety is given in terms of a leaf of an algebraic foliation, and everything is defined over a number field. It turns out that in this case far stronger estimates can be obtained.

### Applying the above to foliations associated to principal G-bundles on various moduli spaces, many classical application of the Pila-Wilkie theorem can be sharpened and effectivized. In particular I will discuss issues around effectivity and polynomial-time solvability for the Andre-Oort conjecture, unlikely intersections in abelian schemes, and some related directions.

Applying the above to foliations associated to principal G-bundles on various moduli spaces, many classical application of the Pila-Wilkie theorem can be sharpened and effectivized. In particular I will discuss issues around effectivity and polynomial-time solvability for the Andre-Oort conjecture, unlikely intersections in abelian schemes, and some related directions.

### Link to recording (110MB)

Link to recording (110MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Sergei Konyagin, *A construction of A. Schinzel - many numbers in a short interval without small prime factors *

(Steklov Institute of Mathematics)

Sergei Konyagin,

*A construction of A. Schinzel - many numbers in a short interval without small prime factors*

(Steklov Institute of Mathematics)

**Thursday, October 22, 2020** *(**8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm** China Standard)*

**Friday, October 23, 2020*** (**2am AEDT, 4am NZDT**)*

**Thursday, October 22, 2020**

*(*

*8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm*

*China Standard)*

**Friday, October 23, 2020**

*(*

*2am AEDT, 4am NZDT*

*)*

### Link to recording (121MB)

Link to recording (121MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Jörg Brüdern, *Harmonic analysis of arithmetic functions*

(University of Göttingen)

Jörg Brüdern,

*Harmonic analysis of arithmetic functions*

(University of Göttingen)

**Tuesday, October 20, 2020** *(**2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm C**hina Standard Time**, 8pm AEDT, 10pm NZDT**)*

**Tuesday, October 20, 2020**

*(*

*2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm C*

*hina Standard Time*

*, 8pm AEDT, 10pm NZDT*

*)*

*Abstract:* We study arithmetic functions that are bounded in mean square, and simultaneously have a mean value over any arithmetic progression. A Besicovitch type norm makes the set of these functions a Banach space. We apply the Hardy-Littlewood (circle) method to analyse this space. This method turns out to be a surprisingly flexible tool for this purpose. We obtain several characterisations of limit periodic functions, correlation formulae, and we give some applications to Waring's problem and related topics. Finally, we direct the theory to the distribution of the arithmetic functions under review in arithmetic progressions, with mean square results of Barban-Davenport-Halberstam type and related asymptotic formulae at the focus of our attention. There is a rich literature on this last theme. Our approach supersedes previous work in various ways, and ultimately provides another characterisation of limit periodic functions: the variance over arithmetic progression is atypically small if and only if the input function is limit periodic.

*Abstract:*We study arithmetic functions that are bounded in mean square, and simultaneously have a mean value over any arithmetic progression. A Besicovitch type norm makes the set of these functions a Banach space. We apply the Hardy-Littlewood (circle) method to analyse this space. This method turns out to be a surprisingly flexible tool for this purpose. We obtain several characterisations of limit periodic functions, correlation formulae, and we give some applications to Waring's problem and related topics. Finally, we direct the theory to the distribution of the arithmetic functions under review in arithmetic progressions, with mean square results of Barban-Davenport-Halberstam type and related asymptotic formulae at the focus of our attention. There is a rich literature on this last theme. Our approach supersedes previous work in various ways, and ultimately provides another characterisation of limit periodic functions: the variance over arithmetic progression is atypically small if and only if the input function is limit periodic.

## Cameron L. Stewart, *On integers represented by binary forms*

(University of Waterloo)

Cameron L. Stewart,

*On integers represented by binary forms*

(University of Waterloo)

**Thursday, October 15, 2020** *(**8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm** China Standard Time)*

**Friday, October 16, 2020*** (**2am AEDT, 4am NZDT**)*

**Thursday, October 15, 2020**

*(*

*8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm*

*China Standard Time)*

**Friday, October 16, 2020**

*(*

*2am AEDT, 4am NZDT*

*)*

*Abstract:* We shall discuss the following results which are joint work with Stanley Xiao.

*Abstract:*We shall discuss the following results which are joint work with Stanley Xiao.

### Let F(x,y) be a binary form with integer coefficients, degree d(>2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z^(2/d).

Let F(x,y) be a binary form with integer coefficients, degree d(>2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z^(2/d).

### Let k be an integer with k>1 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z^(2/d).

Let k be an integer with k>1 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z^(2/d).

### Link to recording (118MB)

Link to recording (118MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Alexander Gorodnik, *Arithmetic approach to the spectral gap problem*

(University of Zurich)

Alexander Gorodnik,

*Arithmetic approach to the spectral gap problem*

(University of Zurich)

**Tuesday, October 13, 2020** *(**2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm C**hina Standard Time**, 8pm AEDT, 10pm NZDT**)*

**Tuesday, October 13, 2020**

*(*

*2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm C*

*hina Standard Time*

*, 8pm AEDT, 10pm NZDT*

*)*

*Abstract:* The spectral gap is an analytic property of group actions which can be described as absence of "almost invariant vectors" or more quantitatively in terms of norm bounds for suitable averaging operators. In the setting of homogeneous spaces this property also has a profound number-theoretic meaning since it is closely related to understanding the automorphic representations. In this talk we survey some previous results about the spectral gap property and describe new approaches to deriving upper and lower bounds for the spectral gap.

*Abstract:*The spectral gap is an analytic property of group actions which can be described as absence of "almost invariant vectors" or more quantitatively in terms of norm bounds for suitable averaging operators. In the setting of homogeneous spaces this property also has a profound number-theoretic meaning since it is closely related to understanding the automorphic representations. In this talk we survey some previous results about the spectral gap property and describe new approaches to deriving upper and lower bounds for the spectral gap.

### Link to recording (112MB)

Link to recording (112MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Philippe Michel, *Simultaneous reductions of CM elliptic curves *

(EPFL)

Philippe Michel,

*Simultaneous reductions of CM elliptic curves*

(EPFL)

**Thursday, October 8, 2020** *(**8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm C**hina Standard Time)*

**Friday, October 9, 2020*** (**2am AEDT, 4am NZDT**)*

**Thursday, October 8, 2020**

*(*

*8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm C*

*hina Standard Time)*

**Friday, October 9, 2020**

*(*

*2am AEDT, 4am NZDT*

*)*

*Abstract:* Let $E$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of discriminant $D<0$. Given $p$ a prime ; if $p$ is inert or ramified in the quadratic field generated by $\sqrt D$ then $E$ has supersingular reduction at a(ny) fixed place above $p$. By a variant of Duke’s equidistribution theorem, as $D$ grows along such discriminants, the proportion of CM elliptic curves with CM by $O_D$ whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes $p_1,\cdots,p_s$ and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank $2$ actions on products of locally homogeneous spaces implies that as $D$ grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual $p_i$ (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement -- obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.

*Abstract:*Let $E$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of discriminant $D<0$. Given $p$ a prime ; if $p$ is inert or ramified in the quadratic field generated by $\sqrt D$ then $E$ has supersingular reduction at a(ny) fixed place above $p$. By a variant of Duke’s equidistribution theorem, as $D$ grows along such discriminants, the proportion of CM elliptic curves with CM by $O_D$ whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes $p_1,\cdots,p_s$ and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank $2$ actions on products of locally homogeneous spaces implies that as $D$ grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual $p_i$ (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement -- obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.

### Link to recording (110MB)

Link to recording (110MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Alexander Smith, *Selmer groups and a Cassels-Tate pairing for finite Galois modules*

(Harvard University)

Alexander Smith,

*Selmer groups and a Cassels-Tate pairing for finite Galois modules*

(Harvard University)

**Monday, October 5, 2020 ****(***5pm PDT, 8pm EDT)*

**Tuesday, October 6, 2020 ****(***1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am C**hina Standard Time**, 11am AEDT, 1pm NZDT**)*

**Monday, October 5, 2020**

**(***5pm PDT, 8pm EDT)*

**Tuesday, October 6, 2020**

**(***1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am C*

*hina Standard Time*

*, 11am AEDT, 1pm NZDT*

*)*

*Abstract:* I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.

*Abstract:*I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.

### As one application of this theory, I will prove an elementary result on the symmetry of the class group pairing for number fields with many roots of unity and connect this to the work of mine and others on class group statistics.

As one application of this theory, I will prove an elementary result on the symmetry of the class group pairing for number fields with many roots of unity and connect this to the work of mine and others on class group statistics.

### This work is joint with Adam Morgan.

This work is joint with Adam Morgan.

### Link to recording (72MB)

Link to recording (72MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Wei Ho, *The Hasse local-to-global principle for some genus one curves*

(University of Michigan)

Wei Ho,

*The Hasse local-to-global principle for some genus one curves*

(University of Michigan)

**Thursday, October 1, 2020** *(**8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm C**hina Standard Time)*

**Friday, October 2, 2020*** (**1am AEST, 4am NZDT**)*

**Thursday, October 1, 2020**

*(*

*8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm C*

*hina Standard Time)*

**Friday, October 2, 2020**

*(*

*1am AEST, 4am NZDT*

*)*

*Abstract:* The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails.

*Abstract:*The Hasse principle is a useful guiding philosophy in arithmetic geometry that relates "global" questions to analogous "local" questions, which are often easier to understand. A simple incarnation of the Hasse principle says that a given polynomial equation has a solution in the rational numbers (i.e., is "globally soluble") if and only if it has a solution in the real numbers and in the p-adic numbers for all primes p (i.e., is "everywhere locally soluble"). While this principle holds for many "simple" such polynomials, it is a very difficult question to classify the polynomials (or more generally, algebraic varieties) for which the principle holds or fails.

### In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, with a special focus on genus one curves given by bihomogeneous polynomials of bidegree (2,2) in \mathbb{P}^1 \times \mathbb{P}^1. For example, we will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves, by comparing the average sizes of 2- and 3-Selmer groups for a family of elliptic curves with a marked point (joint work with Manjul Bhargava).

In this talk, we will discuss problems related to the Hasse principle for some classes of varieties, with a special focus on genus one curves given by bihomogeneous polynomials of bidegree (2,2) in \mathbb{P}^1 \times \mathbb{P}^1. For example, we will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we will explain why the Hasse principle fails for a positive proportion of these curves, by comparing the average sizes of 2- and 3-Selmer groups for a family of elliptic curves with a marked point (joint work with Manjul Bhargava).

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Julie Tzu-Yueh Wang, *Pisot's d-th root's conjecture for function fields and its complex analog*

(Academia Sinica, Taiwan)

Julie Tzu-Yueh Wang,

*Pisot's d-th root's conjecture for function fields and its complex analog*

(Academia Sinica, Taiwan)

**Monday, September 28, 2020 ****(***5pm PDT, 8pm EDT)*

**Tuesday, September 29, 2020 ****(***1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 1pm NZDT)*

**Monday, September 28, 2020**

**(***5pm PDT, 8pm EDT)*

**Tuesday, September 29, 2020**

**(***1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 1pm NZDT)*

### Link to recording (53MB)

Link to recording (53MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Emmanuel Breuillard, *A subspace theorem for manifolds** *

(University of Cambridge)

Emmanuel Breuillard,

*A subspace theorem for manifolds*

*(University of Cambridge)*

**Thursday, September 24, 2020** *(**8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm** China Standard Time)*

**Friday, ****September 25****, 2020*** (1am AEST, 3am NZST)*

**Thursday, September 24, 2020**

*(*

*8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm*

*China Standard Time)*

**Friday,**

**September 25**

**, 2020**

*(1am AEST, 3am NZST)*

*Abstract:* In the late 90's Kleinbock and Margulis solved a long-standing conjecture due to Sprindzuk regarding diophantine approximation on submanifolds of R^n. Their method used homogeneous dynamics via the so-called non-divergence estimates for unipotent flows on the space of lattices. In this talk I will explain how these ideas, combined with a certain understanding of the geometry at the heart of Schmidt's subspace theorem, in particular the notion of Harder-Narasimhan filtration, leads to a metric version of the subspace theorem, where the linear forms are allowed to depend on a parameter. This subspace theorem for manifolds allows to quickly compute certain diophantine exponents, and it leads to several generalizations of the Kleinbock-Margulis results in a variety of contexts. Joint work with Nicolas de Saxcé.

*Abstract:*In the late 90's Kleinbock and Margulis solved a long-standing conjecture due to Sprindzuk regarding diophantine approximation on submanifolds of R^n. Their method used homogeneous dynamics via the so-called non-divergence estimates for unipotent flows on the space of lattices. In this talk I will explain how these ideas, combined with a certain understanding of the geometry at the heart of Schmidt's subspace theorem, in particular the notion of Harder-Narasimhan filtration, leads to a metric version of the subspace theorem, where the linear forms are allowed to depend on a parameter. This subspace theorem for manifolds allows to quickly compute certain diophantine exponents, and it leads to several generalizations of the Kleinbock-Margulis results in a variety of contexts. Joint work with Nicolas de Saxcé.

### Link to recording (176MB)

Link to recording (176MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Ilya D. Shkredov, *Zaremba's conjecture and growth in groups*

(Steklov Mathematical Institute, Moscow)

Ilya D. Shkredov,

*Zaremba's conjecture and growth in groups*

(Steklov Mathematical Institute, Moscow)

**Tuesday, September 22, 2020** *(2am PDT, 5am EDT, 10am BST, **1**1**am CEST**, 12pm IDT, 2:30pm IST, 5**pm **China Standard Time, 7**pm AE**S**T**, 9**pm NZST**)*

**Tuesday, September 22, 2020**

*(2am PDT, 5am EDT, 10am BST,*

*1*

*1*

*am CEST*

*, 12pm IDT, 2:30pm IST, 5*

*pm*

*China Standard Time, 7*

*pm AE*

*S*

*T*

*, 9*

*pm NZST*

*)*

*Abstract:* Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a<q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.

*Abstract:*Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a<q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place for the so-called modular form of Zaremba's conjecture.

### Link to recording (70MB)

Link to recording (70MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Özlem Imamoglu, *A class number formula of Hurwitz*

(ETH Zürich)

Özlem Imamoglu,

*A class number formula of Hurwitz*

(ETH Zürich)

**Thursday, September 17, 2020** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday, ****September 18****, 2020*** (1am AEST, 3am NZST)*

**Thursday, September 17, 2020**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday,**

**September 18**

**, 2020**

*(1am AEST, 3am NZST)*

*Abstract:* In a little known paper Hurwitz gave an infinite series representation for the class number of positive definite binary quadratic forms In this talk I will report on joint work with W. Duke and A. Toth where we show how the ideas of Hurwitz can be applied in other settings, in particular to give a formula for the class number of binary cubic forms.

*Abstract:*In a little known paper Hurwitz gave an infinite series representation for the class number of positive definite binary quadratic forms In this talk I will report on joint work with W. Duke and A. Toth where we show how the ideas of Hurwitz can be applied in other settings, in particular to give a formula for the class number of binary cubic forms.

### Link to recording (89MB)

Link to recording (89MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Dzmitry Badziahin, *Approximation by algebraic numbers*

(University of Sydney)

Dzmitry Badziahin,

*Approximation by algebraic numbers*

(University of Sydney)

**Tuesday, September 15, 2020** *(2am PDT, 5am EDT, 10am BST, **1**1**am CEST**, 12pm IDT, 2:30pm IST, 5**pm **China Standard Time, 7**pm AE**S**T**, 9**pm NZST**)*

**Tuesday, September 15, 2020**

*(2am PDT, 5am EDT, 10am BST,*

*1*

*1*

*am CEST*

*, 12pm IDT, 2:30pm IST, 5*

*pm*

*China Standard Time, 7*

*pm AE*

*S*

*T*

*, 9*

*pm NZST*

*)*

*Abstract:* In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number x, by w_n^*(x) we define the supremum of all positive real values w such that the inequality

*Abstract:*In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number x, by w_n^*(x) we define the supremum of all positive real values w such that the inequality

### |x - a| < H(a)^{-w-1}

|x - a| < H(a)^{-w-1}

### has infinitely many solutions in algebraic real numbers a of degree at most n. Here H(a) means the naive height of the minimal polynomial in Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w_n^*(x) is at least n for all transcendental x? Apart from partial results for small values of n, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w_n^*(x) \ge n/2+1 - o(1). Until recently, the only improvements to this bound were in terms of O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n). More precisely, we show that w_n^*(x) > n/\sqrt{3}.

has infinitely many solutions in algebraic real numbers a of degree at most n. Here H(a) means the naive height of the minimal polynomial in Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w_n^*(x) is at least n for all transcendental x? Apart from partial results for small values of n, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w_n^*(x) \ge n/2+1 - o(1). Until recently, the only improvements to this bound were in terms of O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n). More precisely, we show that w_n^*(x) > n/\sqrt{3}.

### Link to recording (100MB)

Link to recording (100MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Bianca Viray, *Existence of quadratic points on intersections of quadrics*

(University of Washington)

Bianca Viray,

*Existence of quadratic points on intersections of quadrics*

(University of Washington)

**Thursday, September 10, 2020** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday, ****September 11****, 2020*** (1am AEST, 3am NZST)*

**Thursday, September 10, 2020**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday,**

**September 11**

**, 2020**

*(1am AEST, 3am NZST)*

*Abstract:* Springer's theorem and the Amer-Brumer theorem together imply that intersections of two quadrics have a rational point if and only if they have a 0-cycle of degree 1. In this talk, we consider whether this statement can be strengthened in the case when there is no rational point, namely whether 1) the least degree of a 0-cycle can be bounded, and 2) whether there is an effective 0-cycle of this degree. We report on results in this direction, paying particular attention to the case of local and global fields. This is joint work with Brendan Creutz.

*Abstract:*Springer's theorem and the Amer-Brumer theorem together imply that intersections of two quadrics have a rational point if and only if they have a 0-cycle of degree 1. In this talk, we consider whether this statement can be strengthened in the case when there is no rational point, namely whether 1) the least degree of a 0-cycle can be bounded, and 2) whether there is an effective 0-cycle of this degree. We report on results in this direction, paying particular attention to the case of local and global fields. This is joint work with Brendan Creutz.

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Maryna Viazovska, *Universal optimality, Fourier interpolation, and modular integrals*

(EPFL)

Maryna Viazovska,

*Universal optimality, Fourier interpolation, and modular integrals*

(EPFL)

**Tuesday, September 8, 2020** *(1am PDT, 4am EDT, 9am BST, **1**0**am CEST**, 11pm IDT, 1:30pm IST, 4**pm **China Standard Time, 6**pm AE**S**T**, 8**pm NZST**)*

**Tuesday, September 8, 2020**

*(1am PDT, 4am EDT, 9am BST,*

*1*

*0*

*am CEST*

*, 11pm IDT, 1:30pm IST, 4*

*pm*

*China Standard Time, 6*

*pm AE*

*S*

*T*

*, 8*

*pm NZST*

*)*

*Abstract:* In this lecture we will show that the E8 and Leech lattices minimize energy for a wide class of potential functions. This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. Construction of the optimal auxiliary functions attaining these bounds is based on a new interpolation theorem. This is joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko.

*Abstract:*In this lecture we will show that the E8 and Leech lattices minimize energy for a wide class of potential functions. This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. Construction of the optimal auxiliary functions attaining these bounds is based on a new interpolation theorem. This is joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko.

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Kevin Ford, *Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros*

(University of Illinois at Urbana-Champaign)

Kevin Ford,

*Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros*

(University of Illinois at Urbana-Champaign)

**Thursday, September 3, 2020** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday, ****September 4****, 2020*** (1am AEST, 3am NZST)*

**Thursday, September 3, 2020**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday,**

**September 4**

**, 2020**

*(1am AEST, 3am NZST)*

*Abstract:* Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that *any* sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.

*Abstract:*Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that

*any*sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.

### Link to recording (208MB)

Link to recording (208MB)

## Umberto Zannier, *Torsion in elliptic familes and applications to billiards*

(Scuola Normale Superiore Pisa)

Umberto Zannier,

*Torsion in elliptic familes and applications to billiards*

(Scuola Normale Superiore Pisa)

**Tuesday, September 1, 2020** *(2am PDT, 5am EDT, 10am BST, **1**1**am CEST**, 12pm IDT, 2:30pm IST, 5**pm **China Standard Time, 7**pm AE**S**T**, 9**pm NZST**)*

**Tuesday, September 1, 2020**

*(2am PDT, 5am EDT, 10am BST,*

*1*

*1*

*am CEST*

*, 12pm IDT, 2:30pm IST, 5*

*pm*

*China Standard Time, 7*

*pm AE*

*S*

*T*

*, 9*

*pm NZST*

*)*

*Abstract:* We shall consider elliptic pencils, of which the best-known example is probably the Legendre family L_t: y^2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on L_t depending on t) it is an issue to study the set of complex b such that P(b) is torsion on L_b. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,\pi), there are only finitely many cases for which both billiard trajectories are periodic.

*Abstract:*We shall consider elliptic pencils, of which the best-known example is probably the Legendre family L_t: y^2=x(x-1)(x-t) where t is a parameter. Given a section P(t) (i.e. a family of points on L_t depending on t) it is an issue to study the set of complex b such that P(b) is torsion on L_b. We shall recall a number of results on the nature of this set. Then we shall present some applications (obtained jointly with P. Corvaja) to elliptical billiards. For instance, if two players hit the same ball with directions forming a given angle in (0,\pi), there are only finitely many cases for which both billiard trajectories are periodic.

### Link to recording (104MB)

Link to recording (104MB)

### Link to lecture notes (PDF)

Link to lecture notes (PDF)

## Hector Pasten, *A Chabauty-Coleman bound for hyperbolic surfaces in abelian threefolds*

(Pontificia Universidad Católica de Chile)

Hector Pasten,

*A Chabauty-Coleman bound for hyperbolic surfaces in abelian threefolds*

(Pontificia Universidad Católica de Chile)

**Thursday, August 27, 2020** *(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday, ****August 28****, 2020*** (1am AEST, 3am NZST)*

**Thursday, August 27, 2020**

*(8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)*

**Friday,**

**August 28**

**, 2020**

*(1am AEST, 3am NZST)*

*Abstract:* A celebrated result of Coleman gives a completely explicit version of Chabauty's finiteness theorem for rational points in hyperbolic curves over a number field, by a study of zeros of p-adic analytic functions. After several developments around this result, the problem of proving an analogous explicit bound for higher dimensional subvarieties of abelian varieties remains elusive. In this talk I'll sketch the proof of such a bound for hyperbolic surfaces contained in abelian threefolds. This is joint work with Jerson Caro.

*Abstract:*A celebrated result of Coleman gives a completely explicit version of Chabauty's finiteness theorem for rational points in hyperbolic curves over a number field, by a study of zeros of p-adic analytic functions. After several developments around this result, the problem of proving an analogous explicit bound for higher dimensional subvarieties of abelian varieties remains elusive. In this talk I'll sketch the proof of such a bound for hyperbolic surfaces contained in abelian threefolds. This is joint work with Jerson Caro.

### Link to recording (119MB)

Link to recording (119MB)