Curvature-dimension bounds for continuous and discrete spaces

Matthias Erbar

(Universität Bonn)

Event: ERC Workshop on Optimal Transportation and Applications

Date: Oct 30, 2014, time: 09:55

Abstract. Metric measure spaces with curvature-dimension bounds in the sense of Lott-
Villani-Sturm are known to satisfy are number of strong properties in terms of geometric
and functional inequalities. In this talk we first present a new and more simple curvature-
dimension condition based solely on convexity properties of the Shannon entropy. In
combination with linearity of the heat flow it allows to establish a Bochner formula as
the starting point for a much finer analysis on metric measure spaces. Then we make the
connection to the setting of discrete spaces and Markov chains where a notion of lower
curvature bound in the spirit of Lott-Villani-Sturm has been developed recently. The
same new curvature-dimension condition comes as a natural reenforcement of this notion
and allows to give a new notion of dimension for discrete spaces. We will discuss first
consequences of this notion as well as new results on Ricci bounds for discrete spaces. Joint work with Kazumasa Kuwada, Theo Sturm and work in progress with Chris Henderson, Jan Maas, Georg Menz and Prasad Tetali.