Math 858G
Optimal Transport and Geometry

Course Description

In a nutshell, Optimal Transport is concerned with how to move a set of goods or particles from one configuration to another configuration in the most "cost effective" way. Simple as it may sound, it has applications in a surprising number of areas of mathematics. The foundations of the subject was laid by Gaspard Monge in the 1780s and Leonid Kantorovich in the 1940s. While Monge was an applied geometer and Kantorovich made his most important contributions in economics (where he later received the nobel prize), optimal transport itself has since then evolved in several directions and is now an important tool in partial differential equations, probability theory and geometry. This course will give an introduction to optimal transport, covering Monge's and Kantorovich's formulations of the optimal transport problem, basic existence theory and the connection to real Monge-Ampère equations and then move into topics which lie at the intersection of optimal transport and geometry. The content in the later part of the course will depend on the interests of the students and may include:

Probabilistic and dynamic approaches to Kähler-Einstein metrics

Optimal transport and Ricci flow

Curvature obstructions to regularity of optimal transport maps

Synthetic treatment of Ricci curvature

The Sinkhorn algorithm and analogies in Kähler geometry

Optimal Transport and calibrations in pseudo Riemannian geometry

Rough plan

We will dedicate the first four weeks to basic optimal transport and then move into applications. In the later part of the course we will transition from text book material to contemporary research papers. 

Examination/Assignments (preliminary)

During the first four weeks there will be weekly graded homework. For the later part of the course you will be asked to write a number of MathSciNet style reviews of papers and hold at least one presentation of a paper. Active participation during lectures will be strongly encouraged.