This year's lecture was devoted to a brief introduction to stochastic homogenization. In very general and somewhat vague terms, the aim of homogenization theory is to model and mathematically analyze media that are highly heterogeneous due to the presence of " micro-structures ", in an attempt to obtain a macroscopic representation " average ". Possible applications range from complex materials (e.g. composites) to geological media (e.g. porous media). Of course, such an approach cannot succeed without assumptions about the spatial organization (or distribution) of these " micro-structures ". Mathematically, the most studied situation corresponds to a periodic or even almost periodic organization. However, it is more realistic to assume that the distribution is random and stationary (i.e. that " the probability law " is invariant to translation...).
What's more, the periodic (or almost periodic) case is a (very) special case of the stochastic case. And we'll see that, in this case, there are still many open questions.
Lecture