Seminars Sorted by Series

We will describe a recent result on the stability of twisting in 2d Hamiltonian flows and its application to various questions concerning the long-time behavior of 2d Euler solutions. We will look at stability properties of stationary solutions...

I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song...

A fractal uncertainty principle (FUP) roughly says that a 
function and its Fourier transform cannot both be concentrated on a 
fractal set. These were introduced to harmonic analysis in order to 
prove new results in quantum chaos: if eigenfunctions...

I will describe some new "coarse-graining" methods in quantitative homogenization and how they can be used to give rigorous versions of certain heuristic "renormalization group" arguments in physics, with a focus on several examples.

I will present joint work with Jacek Jendrej. We consider classical scalar fields in dimension 1+1 with a symmetric double-well self-interaction potential. Examples of such equations are the phi-4 model and the sine-Gordon equation. These nonlinear...

Hilbert’s sixth problem asks for a mathematically rigorous justification of the macroscopic laws of statistical physics from the microscopic laws of dynamics. The classical setting of this problem is the justification of Boltzmann’s kinetic equation...

We study multiply warped product geometries MN:=Bn×Fn1×···×FnA 

g = g_B + \sum_{a=1}^A v_a^2 g_{F^{n_a}} and show that for an open set of initial data within multiply warped product geometries the Ricci flow starting at any of those develops...

We consider an energy model for N ordered elastic membranes subject to forcing and boundary conditions. The heights of the membranes are described by real functions u_1, u_2,...,u_N, which minimize an energy functional involving the Dirichlet...

A top degree differential form \omega on a smooth algebraic variety X over a local field K gives rise to a (real valued) measure on X(K). The Serre duality yields a natural isomorphism between 1-dimensional spaces of global top degree forms on an...

Modular forms are degree zero cohomology of certain invertible sheaves on modular curves. One is often led to consider also higher cohomology of automorphic vector bundles on Shimura varieties. We try to define and understand the integral coherent...

The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field...

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture...

Tate reformulated the theory of the Riemann zeta function and its functional equation as the Mellin shadow of the Fourier transform on a certain space of function on the adeles. Conjecturally, Langlands' general automorphic L-functions and their...

The linearization method of Dani-Margulis controls the amount of time a unipotent trajectory spends near invariant subvarieties of a homogeneous space. I will describe a problem in number theory where a similar control is desired for diagonalizable...

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation $\pi_0$ at infinity and level dividing some finite upper bound. We compute statistics of this family as the...

I will attempt to give a gentle introduction to the categorical p-adic Langlands program and its connections to questions about congruences between modular forms.

Given a set of integers, we wish to know how many primes there are in the set.  Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information...

I will report on a series of joint works with Gehrmann, Guitart and Masdeu about a plectic generalization of Darmon's Stark-Heegner (SH) points. 

We constructed plectic SH points as p-adic points on elliptic curves and we expect them to control the...

The circle method has been a versatile tool in the study of rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to information about moduli spaces of rational...

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed...

Orbits of many coregular representations of algebraic groups are closely linked to moduli spaces of genus one curves with extra data. We may use these orbit parametrizations to compute the average size of Selmer groups of elliptic curves in certain...

We prove that at least 2/21 of all integers can be written as a sum of two rational cubes, and at least 1/6 of all integers cannot. More generally, in any cubic twist family of elliptic curves, at least one 1/6 of curves have rank 0 and at least 1/6...

Let p be a prime. I want to explain how to use the geometry of modular curves at infinite level and the Hodge–Tate period map to study regular de Rham p-adic Galois representations appearing in the p-adically completed cohomology of modular curves...

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an...

I discuss the spectral and arithmetic side of the relative trace formula of Kuznetsov type for congruence subgroups of SL(n, Z) with applications to automorphic density theorems. A particular focus is on properties of general Kloosterman sums as...

In 2017, Radchenko-Viazovska proved a remarkable interpolation result for even Schwartz functions on the real line: such a function is entirely determined by its values and those of its Fourier transform at square roots of integers. We give a new...

I will talk about a program aimed at exploiting randomness of certain "graphs of symbols" in determining the precise structure of the pro-nilpotent Galois groups of number fields, and how this has been successfully implemented in nilpotency class 2...

Given an automorphic representation \pi of SO(n,n+1) with certain nice properties at infinity, one can nowadays attach to \pi a p-adic Galois representation R of dimension 2n. The Bloch--Kato conjectures then predict in particular that if the L...

We discuss the integrality of theta liftings of anti-cyclotomic characters to a definite unitary group of two variables. This will allow us to construct a Hida family of the theta liftings and relate the congruence module of which to an anti...

We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral.  There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for...

In joint work with Denis Benois, we give an étale construction of Bellaïche's p-adic L-functions about θ-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of p...

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland...

Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture...

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this...

The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in...

Symmetric power functoriality is one of the basic cases of Langlands' functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over Q, as the...