Math 3015 (Topics in Optimal Transport).
Spring 2010

Instructor: Reza Pakzad Email: pakzad At-Sign pitt Dot edu

Lectures: MWF 2:00PM-2:50PM at THACK 524

Outline: Topics in Optimal Transport

Optimal Transport has been a vivid domain of mathematical research in the recent years. Historically, it originated by the Monge problem (1781) which is to minimize the cost of transporting a given amount of homogeneous material mass from the given set of origins to the given set of destinations. Kantorovitch reformulated the problem in the forties by taking a somewhat more modern approach which appears more natural from a theoretical perspective. The Monge-Kantorovitch problem has been a classical subject in probability theory, economics and optimization. Most recently, a new interest arose in the subject, specially due to the pioneering work of Brenier on polar decomposition of space deformations. This lead to an interesting interplay between various fields such as PDEs, fluid mechanics, geometry, probability theory and functional analysis, focused on Optimal Transport as the common denominator.

In this course, we will begin by discussing Monge and Kantorovitch's problems in detail and will cover several proofs of the existence of the optimal plan under different properties of the cost function. The relationship with Kantorovitch's duality theorem (including a proof of the later) will be explored. We will then focus on Brenier's polar decomposition theorem and will try to study it from different perspectives. The Monge-Amp\'ere equation will also be studied as it appears naturally in this context. Finally, if time permits, we will present and study the so called Wasserstein distances. This concept arises naturally in finding the cost of the optimal transport plan based on the problem data but its importance in modern mathematics is not limited to this aspect. In particular we hope to explore relationships of this subject with some topics in Geometric Analysis, e.g. convergence of metric measure spaces and/or isoperimetric inequalities.

The main prerequisite for the course is to have basic background and knowledge in analysis, specially about measure theory and metric spaces, and a little bit of functional analysis. Going beyond the basic concepts, we will always try to review the necessary material in class.

Here is a list of references related to the course material:

1. Topics in Optimal Transportation, Cedric Villani, AMS-GSM 58

2. Partial Differential Equations and Monge-Kantorovich Mass Transfer , L.C. Evans

3. Optimal Transportation and Applications, (Editors: Caffarelli and Salsa), Springer LNM 1813

4. Optimal Transport Old and New, Cedric Villani, Springer SCSM 338.

There will be no exams. For the term work, the students will be required to hand in assigned homeworks and present one research paper or topic during the department Analysis seminar.