Speaker
Description
A classical characterization of Sobolev functions relies on absolute continuity along almost every line parallel to the coordinate axes. Over the past decades, this viewpoint has inspired extensive research in the analysis of metric spaces, revealing deep connections between optimal transport, Sobolev calculus, and Ricci curvature.
After a brief review of these ideas, I will present an approximation strategy in optimal transport aimed at characterizing Sobolev functions on spaces satisfying synthetic Ricci curvature lower bounds. The dynamical viewpoint of Wasserstein geometry provides a natural framework for developing such coordinate-free Lagrangian characterizations, requiring only controlled oscillations of functions along transport geodesics. Finally, I will discuss applications to the Sobolev and BV calculus, including the independence of weak gradients from the integrability exponent and the stability of Cheeger energies under Gromov–Hausdorff convergence. This is based on a series of recent works and will be followed by a related talk by F. Renzi.