RTG Workshop on Optimal Transport and Applications to Differential Geometry

Syllabus

Slots 1, 2
Introduction to optimal transport problem and statement and proof of Brenier's theorem (User's Guide 1–18)
Slot 3
Generalization of optimal transport problem to Riemannian manifolds and proof of McCann's theorem (User's Guide 21–25, see also Optimal Transport and Curvature 11–15, also the original paper: R. J. McCann: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001), no. 3, 589–608)
Slot 4
Proof of Geometric inequalities via optimal transport and stability (User's guide 85-87 and also Alessio Figalli, Stability in geometric and functional inequalities, Proceedings of the 6th European Congress of Mathematics)
Slot 5
Introduction to Wasserstein space of probability measures (User's guide 2.1) and heuristic discussion of its Riemannian structure (User's guide 2.3.2 including Benamou-Brenier's formula)
Slots 6,7
Introduction to gradient flows in metric spaces, derivation of heat equation as gradient flow for entropy in Wasserstein space, connection to log Sobolev inequality (User's guide 55–85)
Slot 8
Basic facts about Ricci curvature (Laplacian comparison, Bishop Gromov theorem) and introduction of weak notion of Ricci curvature for metric measure spaces in terms of displacement convexity; proof that the definition is equivalent to standard one for Riemannian manifolds. (User's guide 123–127)
Slot 9
Introduction to Gromov-Hausdorff convergence and proof that weak lower Ricci bounds pass to Gromov-Hausdorff limits (Burago, Burago and Ivanov 3.6–3.7 for GH Convergence, User's guide 127–130, also Optimal Transport and Curvature 17–18)
Slots 10, 11
Heat flow and calculus on metric measure spaces (including various definitions of Sobolev functions, equivalence of the two approaches to the heat flow as gradient flow and possibly the definition of RCD(K,\infty) spaces). Present parts of the paper: L. Ambrosio, N. Gigli, G. Savare. Heat flow and calculus on metric measure spaces with Ricci curvature bounded below — the compact case.
Slot 12
Integrating by parts in metric measure spaces, Laplacian comparison estimates. N. Gigli, S. Mosconi. The Abresch-Gromoll inequality in a non-smooth setting. (preprint)
Slots 13, 14
Connections to Ricci flow. McCann, Topping. Ricci flow, Entropy and Optimal Transportation.

General References