Differential Geometry and Geometric Analysis

This lecture is part of a course organized by Dale Rolfsen.
 

The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map. The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous ---based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal. See preprint at www.math.toronto.edu/mccann/publications

As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. In the past few years we have gained significant new insight into the question of when there is a symplectic embedding of one basic geometric shape (such as a ball or ellipsoid)into another (such as an ellipsoid or torus). After a brief introduction to symplectic geometry, this talk will describe some of this progress, with particular emphasis on results in dimension four.

Alan Weinstein is a Professor of the Graduate School in the Department of Mathematics at the University of California, Berkeley. He was a colleague of Jerry Marsden throughout Jerry’s career at Berkeley, and their joint papers on “Reduction of symplectic manifolds with symmetry” and “The Hamiltonian structure of the Maxwell-Vlasov equations” were fundamental contributions to geometric mechanics.