Spring 2010

** 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.