Abstract:

"Felix Otto’s pioneering work "The geometry of dissipative evolutions" introduced a geometric perspective on the porous medium equation, interpreting it as a gradient flow in the space of absolutely continuous probability measures equipped with the Wasserstein metric. This insight led to the rigorous framework developed by Ambrosio, Gigli, and Savaré, which formalizes Wasserstein gradient flows and extends Otto’s asymptotic estimates to a broader class of dissipative equations. 

This talk is meant to be an overview of the key aspects of this theory, starting with gradient flows in Hilbert spaces as a motivation and heuristics. The presentation will focus on the three necessary ingredients to define Wasserstein gradient flows: the notion of "tangent" to an (absolutely continuous) curve of measures, the displacement convexity of the functional and the "Wasserstein" subdifferential calculus. The aim is to revisit Otto’s estimates from this abstract framework. Time permiting, we will discuss the Benamou-Brenier formula, and its link to the characterization of tangent vectors in the Wasserstein space."