## Geometry Seminar - Abstracts

### Talk

Friday 27 October 2017, 14:30-15:30 in HG00.308

**Jan Maas** (Institute of Science and Technology Austria)

*Gradient flows and entropy inequalities via optimal transport*

### Abstract

At the end of the 1990s it was discovered by Jordan/Kinderlehrer/Otto
that the diffusion equation can be formulated as a gradient flow in the
space of probability measures, where the driving functional is the
Boltzmann-Shannon entropy, and the dissipation mechanism is given by an
optimal transport metric. This result has been the starting point for
striking developments at the interface of analysis, probability theory,
and geometry.
In this talk I will review work from recent years, in which we
introduced new optimal transport metrics that yield gradient flow
descriptions for discrete stochastic dynamics and for dissipative
quantum systems. This allows us to develop a discrete notion of Ricci
curvature, and to obtain sharp rates of convergence to equilibrium in
several examples. The talk is based on joint work with Matthias Erbar
and with Eric Carlen.

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