Previous Talks

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.