Friday, January 07, 2005

Topics Old and New ..

Odd how topics you've heard about have a tendency to come back. This is an abstract from the UM Math Dept Seminar Bulletin.
Analysis Study
Thursday, January 13, 3:00-5:00, 3866 EH

Juha Heinonen

Optimal transport and synthetic treatment of Ricci curvature bounds (after Lott-Villani and Sturm)

We hope to spend a few lectures in the study seminar on recent works by J. Lott and C. Villani, and K-T. Sturm, who have studied the problem how to define analogs of lower Ricci curvature bounds for metric metric measure spaces. The key idea is to use recent advances in optimal transport and so called Wasserstein distance in the space of probability measures on a given metric space.

This sounds great stuff! The last I heard of Wasserstein spaces was from attending a series of lectures ([1], [2]) by N. Ghoussoub, this past May at CNS Summer School '04. The topic concerned a general inequality which models the free energy interactions of gases, but oddly enough a special case allows a new derivation of Sobolev and Gagliardo-Niremberg inequalities. If I remember rightly, his last lecture advocated the further study of the Wasserstein space, because surprisingly it has some natural notion of a tangent bundle!

Amazing stuff, this! I wouldn't have expected this line of thought - to mix arguments of curvature and bounded geometry with notions of optimal transport and the space of probability measures. Maybe this is a very natural and obvious phenomenon, who knows? But this is one of those moments which I'll be happy to be a buffoon, and learn as much as I can!

1 comment:

Kevin said...

Based on Juha's comments today, my guess is that the connections to Ricci Curvature are, well, probably not very obvious (to me, at least). Speaking of old topics showing up again, Bruce Hanson's talk in GFT was alot like a talk Juha gave in the study seminar last semester. I got to talk to Bruce after study seminar today - he seems like a great guy.