Geometric measure theory: quantitative approaches and applications

In this talk, I will begin with a brief overview of the origins of quantitative geometric theory. Next, I will discuss my past contributions in this area, before focusing on current and future problems. In this last part, I will explain certain advances on problems connecting the differentiability of Lipschitz functions, the presence of Poincaré inequalities, and the geometry of the measures in question. All this will be done in a Euclidean context. I will also mention possible extensions to general metric spaces. I will then conclude with certain applications to statistical learning theory.