# Analysis Seminar

#### The Homogenization Commutator

**Speaker:**
Antoine Gloria, Free University of Brussels

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, May 13, 2016, 1 p.m.

**Synopsis:**

As clear from the mechanical point of view and from the very definition of H-convergence, homogenization of elliptic equations in divergence form is the art of averaging fields and fluxes. At the level of the corrector, this takes the following form: large-scale averages of the flux of the corrector are close to the homogenized coefficients times large-scale averages of the field of the corrector. The defect in this relation is what we call the homogenization commutator. This quantity is key to the structure of fluctuations in stochastic homogenization. On the one hand, when properly rescaled, it converges to Gaussian white noise. On the other hand, it characterizes the fluctuations of the (random variable-coefficients) solution operator in a path-wise sense (in the terminology used for thermal noise). This is based on joint works with Mitia Duerinckx (ULB) and Felix Otto (MPI).