Online Seminar Stochastic Homogenization

TopMath Online Seminar with Prof. Antoine Gloria

2 May 2022 16:00 – 18 July 2022 17:30
Plots of the solutions of the Poisson equation on the unit square (with zero Dirichlet boundary conditions and constant forcing term) for coefficients given by a random checkerboard (the color of each square is picked randomly). Each plot represents the solution for a different size s of the squares of the checkerboard. The solutions display oscillations at scale s, as well as random fluctuations (the solutions would be different for other realisations of the random checkerboard). Both these oscillations and fluctuations disappear in the limit when s goes to zero, regime when homogenization takes place.
Prof. Antoine Gloria (Sorbonne Université) offers in collaboration with TopMath an online seminar on "A tour of quantitative stochastic homogenization" every Monday, 16:00 bis 17:30, from 2nd May to 18th July. The event aims at all interested master's and doctoral students and postdocs in mathematics.

Prof. Antoine Gloria (Laboratoire Jacques-Louis Lions, Sorbonne Université) is an expert of stochastic homogenization. After studying in Paris, he moved to the Hausdorff Center of Mathematics in Bonn as a post-doctoral fellow. He was a research scientist at Inria Lille for four years before joining the Université Libre de Bruxelles as a professor in 2012. Since 2017 he is a full professor at Laboratoire Jacques-Louis Lions. 

A tour of quantitative stochastic homogenization

Stochastic homogenization is the study of solutions of PDEs with random and fast-oscillating coefficients. In the regime when the scale of the oscillations vanishes, one can often replace the original equation by an equation with constant and deterministic (called homogenized) coefficients, leading to a drastic reduction of complexity. As a starting point we shall present these classical qualitative results on the prototypical example of linear equations in divergence form. 

The main aim of the course is to present more recent and quantitative results, addressing the question of oscillations (which amounts to quantifying the difference between the solutions of the original equation and of the homogenized equation) and the question of fluctuations (the solution of the original equation does not only oscillate, but it also displays random fluctuations). We will make an important detour on large-scale regularity issues (culminating on annealed Calderon-Zygmund estimates). Towards the end of the course, we will turn to the case of the linear wave equation and of nonlinear elliptic equations.

Program

Lecture 1 (02.05.): Random coefficients, correctors, and a few words on H-convergence

Lecture 2 (09.05.): Compensated compactness and qualitative stochastic homogenization

Lecture 3 (23.05.): Malliavin calculus and quantitative homogenization in dimension 1

Lecture 4 (30.05.): Control of correctors in higher dimension

Lecture 5 (13.06.): Large-scale regularity for random operators

Lecture 6 (20.06.): Annealed Calderon-Zygmund estimates

Lecture 7 (27.06.): The homogenization commutator and the pathwise structure of fluctuations

Lecture 8 (04.07.): Scaling limit of the homogenization commutator

Lecture 9 (11.07.): Quantitative homogenization of the wave equation

Lecture 10 (18.07.): Quantitative homogenization of elliptic monotone operators

Registration

Registration is requested by May 1st, 2022 (by email to topmath (at) ma.tum.de). You will receive the Zoom login data after registration.