Online-Seminar Stochastische Homogenisierung

TopMath-Onlineseminar mit Prof. Antoine Gloria

2. Mai 2022 16:00 – 18. Juli 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) bietet in Zusammenarbeit mit TopMath vom 2. Mai bis 18. Juli 2022 jeweils montags von 16:00 bis 17:30 Uhr ein Onlineseminar zum Thema "A tour of quantitative stochastic homogenization" an. Die Veranstaltung richtet sich an alle interessierten Master-Studierenden, Promovierenden und Postdocs der Mathematik.

Prof. Antoine Gloria (Laboratoire Jacques-Louis Lions, Sorbonne Université) ist Experte im Bereich Stochastische Homogenisierung. Nach seinem Studium der Mathematik in Paris war er Postdoktorand am Hausdorff-Zentrum für Mathematik in Bonn. Anschließend forschte er vier Jahre an der Inria Lille, bevor er 2012 an die Université Libre de Bruxelles berufen wurde. Seit 2017 ist Antoine Gloria Full Professor am Laboratoire Jacques-Louis Lions.

Das Seminar findet auf Englisch statt.

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.

Programm

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

Anmeldung

Bitte melden Sie sich bis zum 1. Mai 2022 an - per E-Mail an topmath (at) ma.tum.de. Die Zoom-Zugangsdaten erhalten Sie im Anschluss.