Analysis Summer School 2021

A three-day summer school with courses by:

I. Gentil

Title: Sobolev inequalities, gradient flows and conformal invariance

Abstract: In this course, we want to explain why a classical Sobolev inequality in a smooth Riemannian manifold can be proved by using the second derivative of an entropy along its gradient flow. First we prove it in a toy model in a finite dimensional space before the general case under a curvature-dimension condition.

The conformal invariance is a way to prove Sobolev inequalities for different spaces. For instance, the sphere, the Euclidean and the hyperbolic spaces are conformal and enjoy an optimal Sobolev inequality. We would like to prove the recent extension of such result to the so-called Caffarelli-Kohn-Nirenberg inequality. Results proved with the collaboration of Louis Dupaigne and Simon Zugmeyer.

R. Bauerschmidt

Title: Dynamics of spin systems and statistical field theories

Abstract: In this minicourse, I will give a short introduction to the study of the stochastic dynamics of spin systems, discrete and continuous, in terms of Log-Sobolev inequalities associated with the dynamics. I will in particular present an approach through renormalisation that I will illustrate in the example of the sine-Gordon model, a two-dimensional Euclidean field theory with periodic potential. Time permitting, I will also illustrate how this approach allows to couple this model to the Gaussian free field.

O. Domínguez & M. Milman

Title: Commutator estimates revisited

Abstract: The boundedness of some of the fundamental operators in Classical Analysis and PDEs depends on subtle cancellations. We review some examples to set the stage for a general treatment using interpolation theory. In the second part of the course we present new methods to construct function spaces and their application to control commutators.

For any queries, please contact one of the organisers: