BZ Seminar in Analysis: Tapio Rajala (Jyväskylä)

Ricci curvature lower bounds and branching geodesics in metric spaces

In metric measure spaces a notion of Ricci curvature lower bounds based on optimal mass transportation were introduced by Sturm and by Lott and Villani. In order to obtain results analogous to the ones in smooth Riemannian case an extra assumption of nonbranching has often been assumed. I will present some recent work related to the nonbranching assumption. I will first discuss how to improve geodesics in the space of probability measures and how to use these geodesics to prove local Poincaré inequalities without assuming nonbranching. After this I will concentrate on the definition by Ambrosio, Gigli and Savaré of Riemannian Ricci curvature lower bounds and its connection to the improved geodesics and nonbranching. Part of the results I plan to present were obtained in collaboration with L. Ambrosio, N. Gigli, A. Mondino and K.-Th. Sturm.

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