March 2, 2018Location: Janskerkhof 3 (Utrecht), room 019
We start by developing basic notions of stochastic differential geometry in the framework of static Riemannian manifolds and manifolds evolving under a geometric flow. We discuss in particular the notion of a canonical Brownian motion in this geometric setting. The results will be used to approach Ricci curvature, respectively Ricci flow, from a probabilistic point of view. The so-called Ricci flow is a prominent example of a geometric flow on a manifold; it may be seen as a kind of nonlinear heat equation on the space of Riemannian metrics on the manifold. It can be used to smoothen out the geometry of the manifold towards the best possible one determined by the topology of the manifold.
In the past decade there has been a lot of progress in the analysis on metric measure spaces based on ideas from optimal transport. In this talk I will present an overview of recent work, which develops some of these ideas in the context of discrete stochastic dynamics and dissipative quantum systems. In particular, I will present a new class of transport metrics between density matrices, which can be used to obtain entropy inequalities and sharp rates of convergence to equilibrium in some interesting models.