A glimpse into the differential topology and geometry of optimal transport

Abstract

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic 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 also 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.