Special Colloquium: Quantitative homogenization, renormalization and anomalous diffusion

I will begin the talk with an overview of the topic of quantitative homogenization for elliptic and parabolic equations. Homogenization refers to the procedure of replacing a very “noisy” equation– one with rapidly oscillating coefficients– with a nicer, “effective” equation in a large-scale limit. There is a very abstract theory of (qualitative) homogenization, which is classical. We will discuss the more concrete theory of quantitative homogenization, which has been developed recently. A central role in the story concerns certain “coarse-graining” arguments, which can be seen as constituting a rigorous renormalization group-type approach, formulated in the language of analysis. These methods have surprising applications in mathematical physics and probability, which are still emerging. In the second part of the talk, I will discuss one such application (in a recent joint work with V. Vicol) to turbulence theory: namely, a proof of anomalous diffusion for an advection-diffusion equation.

Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor