V5F4 Selected Topics in Stochastic Analysis
Introduction to Optimal Transport and Applications
Time: Monday 8 ct
Room: SR 1.008
Description:
The optimal transport problem has a long history dating back to Monge in the 18th century. In modern terms the problem is, given two probability distributions \(\mu,\nu\) on \(\mathbb{R}^n\), to find a transport map T pushing \(\mu\) forward to \(\nu\) that minimizes the transport cost
\[\int c(x,T(x)) \mathrm{d} \mu(x)\]
for a given cost function c.
In the last two decades the theory has received new attention and seen an enormous development. Striking connections to a number of mathematical fields have been established ranging from probability and economics to PDE and Riemannian geometry, where optimal transport is used as a powerful and versatile tool.
In the first part of the lecture we will consider the optimal transport problem in a general setting and cover the beautiful theory leading to existence and characterization of solutions.
The second part of the lecture will give an introduction to some recent applications of optimal transport. Possible topics include:
- Otto's geometric interpretation of evolution PDEs: Heat and porous medium equations e.g. as gradient flows in the space of probability measures, Theory of gradient flows in metric spaces
- Geometry of singular spaces: How to define a notion of Ricci curvature lower bounds for metric spaces equipped with a measure using optimal transport and properties of such spaces
- Using optimal transport to prove geometric and functional inequalities, e.g. isoperimetry and concentration
- Variants of the transport problem
Prerequisites:
A solid background in measure theory is desirable. Some basic knowledge of Riemannian geometry and PDE will be helpful but is not strictly required.
References:
- C. Villani, Topics in optimal transportation
- C. Villani, Optimal transport, old and new
- L. Ambrosio, N. Gigli, A user's guide to optimal transport
- F. Otto, The geometry of dissipative evolution equations: The porous medium equation, article
- K.-Th. Sturm, The geometry of metric measure spaces, article