Online Seminar Stochastic Homogenization
TopMath Online Seminar with Prof. Antoine Gloria
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.
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 is requested by May 1st, 2022 (by email to topmath (at) ma.tum.de). You will receive the Zoom login data after registration.