Previous Talks

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 Israel Standard Time, 3:30pm IST, 6pm China 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.

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.

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)

Abstract: Link to Abstract.

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 China 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.

Link to recording (148MB)
Slides are available by request from the speaker.

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)

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 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).

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 China 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.

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 China 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.

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 China 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.

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.

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 China 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.

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).

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)

Abstract: Link to Abstract.

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)

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.

Ilya D. Shkredov, Zaremba's conjecture and growth in groups
(Steklov Mathematical Institute, Moscow)

Tuesday, September 22, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm 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.

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)

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.

Dzmitry Badziahin, Approximation by algebraic numbers
(University of Sydney)

Tuesday, September 15, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm 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

|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}.

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)

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.

Maryna Viazovska, Universal optimality, Fourier interpolation, and modular integrals
(EPFL)

Tuesday, September 8, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11pm IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm 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.

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)

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.

Umberto Zannier, Torsion in elliptic familes and applications to billiards
(Scuola Normale Superiore Pisa)

Tuesday, September 1, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm 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.

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)

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.

Christopher Skinner, Solving diagonal diophantine equations over general p-adic fields
(Princeton University)

Thursday, August 20, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 21, 2020 (1am AEST, 3am NZST)

Abstract: This talk will explain a proof that a system of r diagonal equations

a_{i,1} x_1^d + ...+ a_{i,s} x_s^d = 0 , i = 1,...,r

with coefficients in a p-adic field K has a non-trivial solution in K if the number of variables s exceeds 3r^2d^2 (if p > 2) or 8r^2d^2 (if p=2). This is the first bound that holds uniformly for all p-adic fields K and that is polynomial in r or d. The methods -- and talk -- are elementary.

Carl Pomerance, "Practical numbers"
(Dartmouth College)

Thursday, August 13, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 14, 2020 (1am AEST, 3am NZST)

Abstract: A practical number n is one where each number up to n can be expressed as a subset sum of n's positive divisors. It seems that Fibonacci was interested in them since they have the property that all fractions m/n with m < n can be written as a sum of distinct unit fractions with denominators dividing n. With similar considerations in mind, Srinivasan in 1948 coined the term "practical". There has been quite a lot of effort to study their distribution, effort which has gone hand in hand with the development of the anatomy of integers. After work of Tenenbaum, Saias, and Weingartner, we now know the "Practical Number Theorem": the number of practical numbers up to x is asymptotically cx/log x, where c= 1.33607.... In this talk I'll discuss some recent developments, including work of Thompson who considered the allied concept of phi-practical numbers n (the polynomial t^n-1 has divisors over the integers of every degree up to n) and the proof (joint with Weingartner) of a conjecture of Margenstern that each large odd number can be expressed as a sum of a prime and a practical number.


Bjorn Poonen, "Tetrahedra with rational dihedral angles"
(MIT)

Thursday, August 6, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, August 7, 2020 (1am AEST, 3am NZST)

Abstract: In 1895, Hill discovered a 1-parameter family of tetrahedra whose dihedral angles are all rational multiples of pi. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity. Many previous authors have solved polynomial equations in roots of unity, but never with more than 12 monomials, and the Conway-Jones polynomial has 105 monomials! I will explain the method we use to solve it and our discovery that the full classification consists of two 1-parameter families and an explicit finite list of sporadic tetrahedra.

Building on this work, we classify all configurations of vectors in R^3 such that the angle between each pair is a rational multiple of pi. Sample result: Ignoring trivial families and scalar multiples, any configuration with more than 9 vectors is contained in a particular 15-vector configuration.

This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein.


Joseph H. Silverman, "More Tips on Keeping Secrets in a Post-Quantum World: Lattice-Based Cryptography"
(Brown University)

Thursday, July 30, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 31, 2020 (1am AEST, 3am NZST)

Abstract: What do internet commerce, online banking, and updates to your phone apps have in common? All of them depend on modern public key cryptography for security. For example, there is the RSA cryptosystem that is used by many internet browsers, and there is the elliptic curve based ECDSA digital signature scheme that is used in many applications, including Bitcoin. All of these cryptographic construction are doomed if/when someone (NSA? Russia?  China?) builds a full-scale operational quantum computer. It hasn't happened yet, as far as we know, but there are vast resources being thrown at the problem, and slow-but-steady progress is being made. So the search is on for cryptographic algorithms that are secure against quantum computers.  The first part of my talk will be a mix of math and history and prognostication centered around the themes of quantum computers and public key cryptography. The second part will discuss cryptographic constructions based on hard lattice problems, which is one of the approaches being proposed to build a post-quantum cryptographic infrastructure.

Jordan Ellenberg, "What’s up in arithmetic statistics?"
(University of Wisconsin–Madison)

Thursday, July 23, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard)
Friday, July 24, 2020 (1am AEST, 3am NZST)

Abstract: If not for a global pandemic, a bunch of mathematicians would have gathered in Germany to talk about what’s going on in the geometry of arithmetic statistics, which I would roughly describe as “methods from arithmetic geometry brought to bear on probabilistic questions about arithmetic objects". What does the maximal unramified extension of a random number field look like? What is the probability that a random elliptic curve has a 2-Selmer group of rank 100? How do you count points on a stack? I’ll give a survey of what’s happening in questions in this area, trying to emphasize open questions.

Wadim Zudilin, "Irrationality through an irrational time"
(Radboud University Nijmegen)

Tuesday, July 21, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: After reviewing some recent development and achievements related to diophantine problems of the values of Riemann's zeta function and generalized polylogarithms (not all coming from myself!), I will move the focus to \pi=3.1415926... and its rational approximations. Specifically, I will discuss a construction of rational approximations to the number that leads to the record irrationality measure of \pi. The talk is based on joint work with Doron Zeilberger.

Jennifer Balakrishnan, "A tale of three curves"
(Boston University)

Thursday, July 16, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 17, 2020 (1am AEST, 3am NZST)

Abstract: We will describe variants of the Chabauty-Coleman method and quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.


Ken Ono, "Variants of Lehmer's speculation for newforms"
(University of Virginia)

Monday, July 13, 2020 (5pm PDT, 8pm EDT)
Tuesday, July 14, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)

Abstract: In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of τ(n), or is a Fourier coefficient of any given newform. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained some results that will be described here. For example, infinitely many spaces are presented for which the primes ℓ≤37 are not absolute values of coefficients of any newforms with integer coefficients. For Ramanujan’s tau-function, such results imply, for n>1, that

τ(n)∉{±1,±3,±5,±7,±13,±17,−19,±23,±37,±691}.

Lillian Pierce, "On Bourgain’s counterexample for the Schrödinger maximal function"
(Duke University)

Thursday, July 9, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 10, 2020 (1am AEST, 3am NZST)

Abstract: There is a long and visible history of applications of analytic methods to number theory. More recently we are starting to recognize applications of number-theoretic methods to analysis. In this talk we will describe an important recent application in this direction.

In 1980, Carleson asked a question in PDE's: for what class of initial data functions does a pointwise a.e. convergence result hold for the solution of the linear Schrödinger equation? Over the next decades, many people developed counterexamples to show “necessary conditions,” and on the other hand positive results to show “sufficient conditions.” In 2016 Bourgain wrote a 3-page paper using facts from number theory to construct a family of counterexamples. A 2019 Annals paper of Du and Zhang then resolved the question by proving positive results that push the “sufficient conditions” to meet Bourgain’s “necessary conditions."

Bourgain’s construction was regarded as somewhat mysterious. In this talk, we give an overview of how to rigorously derive Bourgain’s construction using ideas from number theory. Our strategy is to start from “zero knowledge" and gradually optimize the set-up to arrive at the final counterexample. This talk will be broadly accessible.

René Schoof, "Abelian varieties over Q(\sqrt{97}) with good reduction everywhere"
(Università di Roma “Tor Vergata”)

Tuesday, July 7, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: Under assumption of the Generalized Riemann Hypothesis we show that every abelian variety over Q(\sqrt{97}) with good reduction everywhere is isogenous to a power of a certain 3-dimensional modular abelian variety.

(joint with Lassina Dembele)

James Maynard, "Primes in arithmetic progressions to large moduli"
(University of Oxford)

Thursday, July 2, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, July 3, 2020 (1am AEST, 3am NZST)

Abstract: How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis!

If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.

Kannan Soundararajan, "Equidistribution from the Chinese Remainder Theorem"
(Stanford University)

Monday, June 29, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 30, 2020 (1am BST, 2am CEST, 3am IDT, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)

Abstract: Suppose for each prime p we are given a set A_p (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set A_q of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in A_q will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors. (Joint work with Emmanuel Kowalski.)

Peter Sarnak, "Integer points on affine cubic surfaces"
(Institute for Advanced Study and Princeton University)

Thursday, June 25, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 26, 2020 (1am AEST, 3am NZST)

Abstract: The level sets of a cubic polynomial in four or more variables tends to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in say a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in SL_2 over such rings.

Link to recording (172MB) (technical problem after about 46 minutes)

Igor Shparlinski, "Weyl Sums: Large, Small and Typical"
(UNSW Sydney)

Tuesday, June 23, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 11am IDT, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: While Vinogradov’s Mean Value Theorem, in the form given by J. Bourgain, C. Demeter and L. Guth (2016) and T. Wooley (2016-2019), gives an essentially optimal result on the power moments of the Weyl sums

S(u;N) =\sum_{1\le n \le N} \exp(2 \pi i (u_1n+…+u_dn^d))

where u = (u_1,...,u_d) \in [0,1)^d, very little is known about the distribution, or even existence, of u \in [0,1)^d, for which these sums are very large, or small, or close to their average value N^{1/2}. In this talk, we describe recent progress towards these and some related questions.

We also present some new bounds on S(u;N) which hold for almost all (u_i)_{i\in I} and all (u_j)_{j\in J}, where I \cup J is a partition of {1,…,,d}. These bounds improve similar results of T. Wooley (2015). Our method also applies to binomial sums

T(x,y; N) = \sum_{1\le n \le N} \exp(2 \pi i (xn+yn^d))

with x,y \in [0,1), in which case we improve some results of M.B. Erdogan and G. Shakan (2019).

This is a joint work with Changhao Chen and Bryce Kerr.

Elon Lindenstrauss, "Diagonalizable flows, joinings, and arithmetic applications"
(Hebrew University of Jerusalem)

Thursday, June 18, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, June 19, 2020 (1am AEST, 3am NZST)

Abstract: Rigidity properties of higher rank diagonalizable actions have proved to be powerful tools in understanding the distribution properties of rational tori in arithmetic quotients. Perhaps the simplest, and best known, example of such an equidistribution question is the equidistribution of CM points of a given discriminant on the modular curve. The equidistribution of CM points was established by Duke using analytic methods, but for finer questions (and questions regarding equidistribution on higher rank spaces) the ergodic theoretic approach has proved to be quite powerful.

I will survey some of the results in this direction, including several results about joint distributions of collections of points in product spaces by Aka, Einsiedler, Khayutin, Shapira, Wieser and other researchers.

Harald Andrés Helfgott, "Optimality of the logarithmic upper-bound sieve, with explicit estimates"
(Göttingen/CNRS (IMJ))

Tuesday, June 16, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: At the simplest level, an upper bound sieve of Selberg type is a choice of rho(d), d<=D, with rho(1)=1, such that

S = \sum_{n\leq N} \left(\sum_{d|n} \mu(d) \rho(d)\right)^2

is as small as possible.

The optimal choice of rho(d) for given D was found by Selberg. However, for several applications, it is better to work with functions rho(d) that are scalings of a given continuous or monotonic function eta. The question is then what is the best function eta, and how does S for given eta and D compares to S for Selberg's choice.

The most common choice of eta is that of Barban-Vehov (1968), which gives an S with the same main term as Selberg's S. We show that Barban and Vehov's choice is optimal among all eta, not just (as we knew) when it comes to the main term, but even when it comes to the second-order term, which is negative and which we determine explicitly.

This is joint work with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero.

Yuri Bilu, "Trinomials, singular moduli and Riffaut's conjecture"
(University of Bordeaux)

Thursday, June 11, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 12, 2020 (1am AEST, 3am NZST)

Abstract: Riffaut (2019) conjectured that a singular modulus of degree h>2 cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results. A joint work with Florian Luca and Amalia Pizarro.

Felipe Voloch, "Value sets of sparse polynomials"
(University of Canterbury)

Monday, June 8, 2020 (5pm PDT, 8pm EDT)
Tuesday, June 9, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)

Abstract: We obtain a lower bound on the size of the value set f(F_p) of a sparse polynomial f(x) in F_p[x] over a finite field of p elements when p is prime. This bound is uniform with respect to the degree and depends on the number of terms of f.

Timothy Browning, "Random Diophantine equations"
(IST Austria)

Thursday, June 4, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, June 5, 2020 (1am AEST, 3am NZST)

Abstract: I’ll survey some of the key challenges around the solubility of polynomial Diophantine equations over the integers.

While studying individual equations is often extraordinarily difficult, the situation is more accessible if we merely ask what happens on average and if we restrict to the so-called Fano range, where the number of variables exceeds the degree of the polynomial. Indeed, about 20 years ago, it was conjectured by Poonen and Voloch that random Fano hypersurfaces satisfy the Hasse principle, which is the simplest necessary condition for solubility. After discussing related results I’ll report on joint work with Pierre Le Boudec and Will Sawin where we establish this conjecture for all Fano hypersurfaces, except cubic surfaces.

Kaisa Matomäki, "Multiplicative functions in short intervals revisited"
(University of Turku)

Tuesday, June 2, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: A few years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals [x, x+h] is close to its average on a long interval [x, 2x]. This result has since been utilized in many applications.

In a work in progress that I will talk about, Radziwill and I revisit the problem and generalise our result to functions which vanish often as well as prove a power-saving upper bound for the number of exceptional intervals (i.e. we show that there are O(X/h^\kappa) exceptional x \in [X, 2X]).

We apply this result for instance to studying gaps between norm forms of an arbitrary number field.

Trevor Wooley, "Bracket quadratics, Hua’s Lemma and Vinogradov’s mean value theorem"
(Purdue University)

Thursday, May 28, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 29, 2020 (1am AEST, 3am NZST)

Abstract: A little over a decade ago, Ben Green posed the problem of showing that all large integers are the sum of at most a bounded number of bracket quadratic polynomials of the shape n[n\theta], for natural numbers n, in which \theta is an irrational number such as the square-root of 2. This was resolved in the PhD thesis of Vicky Neale, although no explicit bound was given concerning the number of variables required to achieve success. In this talk we describe a version of Hua’s lemma for this problem that can be applied via the Hardy-Littlewood method to obtain a conclusion with 5 variables. The associated argument differs according to whether \theta is a quadratic irrational or not. We also explain how related versions of Hua’s lemma may be interpreted in terms of discrete restriction variants of Vinogradov’s mean value theorem, thus providing a route to generalisation.

Bas Edixhoven, "Geometric quadratic Chabauty"
(Leiden University)

Tuesday, May 26, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: Joint work with Guido Lido (see arxiv preprint). Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). Our work gives a version of this method that uses only `simple algebraic geometry' (line bundles over the jacobian and models over the integers). For the talk, no knowledge of all this algebraic geometry is required, it will be accessible to all number theorists.

Arizona Winter School 2020: http://swc.math.arizona.edu/index.html

Zeev Rudnick, "Prime lattice points in ovals"
(Tel-Aviv University)

Thursday, May 21, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time)
Friday, May 22, 2020 (1am AEST, 3am NZST)

Abstract: The study of the number of lattice points in dilated regions has a long history, with several outstanding open problems. In this lecture, I will describe a new variant of the problem, in which we study the distribution of lattice points with prime coordinates. We count lattice points in which both coordinates are prime, suitably weighted, which lie in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. We obtain an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term. (joint work with Bingrong Huang).

Kristin Lauter, "How to Keep your Secrets in a Post-Quantum World"
(Microsoft Research Redmond Labs)

Monday, May 18, 2020 (5pm PDT, 8pm EDT)
Tuesday, May 19, 2020 (1am BST, 2am CEST, 5:30am IST, 8am China Standard Time, 10am AEST, 12pm NZST)

Abstract: As we move towards a world which includes quantum computers which exist at scale, we are forced to consider the question of what hard problems in mathematics our next generation of cryptographic systems will be based on. Supersingular Isogeny Graphs were proposed for use in cryptography in 2006 by Charles, Goren, and Lauter. Supersingular Isogeny Graphs are examples of Ramanujan graphs, which are optimal expander graphs. These graphs have the property that relatively short walks on the graph approximate the uniform distribution, and for this reason, walks on expander graphs are often used as a good source of randomness in computer science. But the reason these graphs are important for cryptography is that finding paths in these graphs, i.e. routing, is hard: there are no known subexponential algorithms to solve this problem, either classically or on a quantum computer. For this reason, cryptosystems based on the hardness of problems on Supersingular Isogeny Graphs are currently under consideration for standardization in the NIST Post-Quantum Cryptography (PQC) Competition. This talk will introduce these graphs, the cryptographic applications, and the various algorithmic approaches which have been tried to attack these systems.

Valentin Blomer, "Joint equidistribution and fractional moments of L-functions"
(Universität Bonn)

Thursday, May 14, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)

Abstract: Integral points on spheres of large radius D^(1/2) equidstribute (subject to appropriate congruence conditions), and so do Heegner points of large discriminant D on the modular curve. Both sets have roughly the same cardinality, and there is a natural way to associate with each point on the sphere a Heegner point. Do these pairs equidistribute in the product space of the sphere and the modular curve as D tends to infinity?

A seemingly very different, but structurally similar joint equidistribution problem can be asked for the supersingular reduction at two different primes of elliptic curves with CM by an order of large discriminant D.

Both equidistribution problems have been studied by ergodic methods under certain conditions on D. I will explain how to use number theory and families of high degree L-functions to obtain an effective equidistribution statement with a rate of convergence, assuming GRH. This is joint work in progress with F. Brumley.

Michel Waldschmidt, "Representation of integers by cyclotomic binary forms"
(Sorbonne University)

Tuesday, May 12, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)

Abstract: The representation of positive integers as a sum of two squares is a classical problem studied by Landau and Ramanujan. A similar result has been obtained by Bernays for positive definite binary form. In joint works with Claude Levesque and Etienne Fouvry, we consider the representation of integers by the binary forms which are deduced from the cyclotomic polynomials. One main tool is a recent result of Stewart and Xiao which generalizes the theorem of Bernays to binary forms of higher degree.

Andrew Sutherland, "Sums of three cubes"
(MIT)

Thursday, May 7, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 8:30pm IST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)

Abstract: In 1953 Mordell asked whether one can represent 3 as a sum of three cubes in any way other than 1^3+1^3+1^3 and 4^3+4^3 -5^3. Mordell's question spurred many computational investigations over the years, and while none found a new solution for 3, they eventually determined which of the first 100 positive integers k can be represented as a sum of three cubes in all but one case: k=42.

In this talk I will present joint work with Andrew Booker that used Charity Engine's crowd-sourced compute grid to affirmatively answer Mordell's question, as well as settling the case k=42. I will also discuss a conjecture of Heath-Brown that predicts the existence of infinitely many more solutions and also explains why they are so difficult to find

Andrew Granville, "Frobenius's postage stamp problem, and beyond..."
(Université de Montréal)

Thursday, April 30, 2020 (8am PDT, 11am EDT, 5pm CEST, 11pm China Standard Time, (+1) 1am AEST, (+1) 3am NZST)

Abstract: We study this famous old problem from the modern perspective of additive combinatorics, and then look at generalizations.