Thu, 06 Jun 2024
16:00
L5

Intersections of geodesics on modular curves and Hilbert modular forms

Håvard Damm-Johnsen
( Oxford)
Abstract

The 12th of Hilbert's 23 problems posed in 1900 asks for an explicit description of abelian extensions of a given base field. Over the rationals, this is given by the exponential function, and over imaginary quadratic fields, by meromorphic functions on the complex upper half plane.  Darmon and Vonk's theory of rigid meromorphic cocycles, or "RM theory", includes conjectures giving a $p$-adic solution over real quadratic fields. These turn out to be closely linked to purely topological questions about intersections of geodesics in the upper half plane, and to $p$-adic deformations of Hilbert modular forms. I will explain an extension of results of Darmon, Pozzi and Vonk proving some of these conjectures, and some ongoing work concerning analogous results on Shimura curves.

Thu, 23 May 2024
16:00
L5

Square roots for symplectic L-functions and Reidemeister torsion

Amina Abdurrahman
(IHES)
Abstract

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

Thu, 16 May 2024
16:00
L5

Ergodic Approach to the Mixing Conjecture

George Robinson
( Oxford)
Abstract

The Mixing Conjecture of Michel-Venkatesh has now taken on additional arithmetic significance via Wiles' new approach to modularity. Inspired by this, we present the best currently available method, pioneered by Khayutin's proof for quaternion algebras over the rationals, which we have successfully applied to totally real fields. The talk will overview the method, which brings a suprising combination of ergodic theory, analysis and geometry to bear on this arithmetic problem.

Mon, 17 Feb 2025
16:30
L4

Stable Free Boundaries in Dimension 3: Bernoulli and Allen--Cahn

Xavier Fernandez-Real
(EPFL)
Abstract
In this talk, we present a forthcoming work on the classification of global stable solutions to the Bernoulli problem in $\mathbb R^3$. In particular, this yields local universal curvature bounds for the free boundary for the local problem.
By means of this result, we prove the free boundary Allen--Cahn stability conjecture in dimension 3: global stable solutions to the free boundary analogue of the Allen--Cahn equation are one dimensional in dimension 3. This solves a long-standing conjecture in the free boundary case.

 
Mon, 06 May 2024
16:30
L4

On Galerkin approximations of the 2D Euler equations

Luigi Berselli
(University of Pisa)
Abstract

We study fully discrete approximation of the 2D Euler equations for ideal homogeneous fluids. We focus on spectral methods and  discuss rates of convergence of velocity and vorticity under different assumptions on the smoothness of the data.

Mon, 20 May 2024
15:30
L3

Multiscale analysis of wave propagation in random media

Prof Josselin Garnier
(Centre de Mathematiques Appliquees, Ecole polytechnique, Institut Polytechnique de Paris)
Further Information

This is a joint seminar with the Stochastic Analysis & Mathematical Finance seminar.

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