## An Introduction to Liouville Theory

Antti Kupiainen

University of Helsinki

May 15, 2018

Antti Kupiainen

University of Helsinki

May 15, 2018

Tarek Elgindi

Princeton University

February 22, 2017

We discuss the problem of singularity formation for some of the basic equations of incompressible fluid mechanics such as the incompressible Euler equation and the surface quasi-geostrophic (SQG) equation. We begin by going over some of the classical model equations which have been proposed to understand the dynamics of these equations such as the models of Constantin-Lax-Majda and De Gregorio. We then explain our recent proof of singularity formation in De Gregorio's model.

Eric Carlen

Rutgers University

February 15, 2017

We present joint work with Jan Maas showing that Quantum Markov semigroups satisfying a detailed balance condition are gradient flow for quantum relative entropy, and use this prove some conjectured inequalities arising in quantum information theory.

Christoph Marx

Oberlin College

January 25, 2017

In this talk we will quantify the coupling asymptotics for the Lyapunov exponent (LE) of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. By proving an asymptotic formula for the LE valid for all irrational frequencies, our result refines the well-known lower bound by Sorets and Spencer.

Pierre Tarres

Université Paris-Dauphine

January 24, 2017

We explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism. These correspondences enable us to show recurrence/transience results on the Edge-reinforced random walk, and they also allow us to provide insight into these models.

Sören Petrat

Member, School of Mathematics

November 16, 2016

The talk is about the dynamics of a tracer particle coupled strongly to a dense non-interacting electron gas in two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times.

Douglas Stanford

Member, School of Natural Sciences

October 26, 2016

Percy Deift

New York University

October 19, 2016

This is joint work with Tom Trogdon. Here the author shows how to prove universality rigorously for certain numerical algorithms of the type described in the first lecture. The proofs rely on recent state of the art results from random matrix theory.

Percy Deift

New York University

October 13, 2016

This is joint work with Govind Menon, Sheehan Olver and Thomas Trogdon. The speaker will present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time.

Peter Pickl

Ludwig-Maximilians-Universität München

October 4, 2016

The rigorous derivation of the Vlasov equation from Newtonian mechanics of $N$ Coulomb-interacting particles is still an open problem. In the talk I will present recent results, where an $N$-dependent cutoff is used to make the derivation possible. The cutoff is removed as the particle number goes to zero. Our result holds for typical initial conditions, only. This is, however, not a technical assumption: one can in fact prove deviation from the Vlasov equation for special initial conditions for the system we consider.