Dr. Francesca Anceschi “On the weak regularity theory for the Kolmogorov equation”

The Kolmogorov equation is a strongly degenerate second order pde,
that was firstly introduced in 1934 as a fundamental ingredient for the
description of the density of a system of n particles of gas in the phase
space. Later on, Hörmander considered it as a prototype for the family
of hypoelliptic operators that can be written as a sum of squares.
Nowadays, the research community aims to study the weak regularity
theory for this class of equations, by extending to this hypoelliptic
framework the results that hold true in the elliptic and parabolic setting.
However, due to the strong degeneracy of the equation various
precautions are required. In this talk, we introduce the geometrical
framework suitable for the study of the family of Kolmogorov equations
and we present the ideas behind the most recent developments in the
study of the weak regularity theory for this class of equations.