# Course Identification

Optimal transport and convex localization
20234142

## Lecturers and Teaching Assistants

Prof. Boaz Binyamin Klartag
N/A

## Course Schedule and Location

2023
Second Semester
Wednesday, 13:15 - 16:00, Ziskind, Rm 155
19/04/2023
28/06/2023

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points

9 lectures course.

## Prerequisites

Familiarity with basic Measure Theory (say, Lebesgue measure) and Differential Geometry (say, smooth manifold).

100

English

## Attendance and participation

N/A

Numerical (out of 100)

100%

Final assignment

N/A
N/A
-
N/A

N/A

## Syllabus

What is the optimal way to move given piles of sand in order to fill up given holes of the same total volume, so as to minimize the work done? This question, going back to Monge in the 18th century, is the starting point of the theory of Optimal Transport.

In this class we will explore various methods for transport, rearrangement and decomposition of mass. We will then use these methods in order to prove geometric and functional inequalities such as isoperimetric inequalities, Poincare inequalities, Log-Sobolev inequalities and Brunn-Minkowski type inequalities. These inequalities are used, in turn, for establishing Concentration of Measure estimates in high dimensions.

In the first part of the course we will study mass transport and its applications in Euclidean spaces and spheres. We will study the Brenier map as well as Convex localization techniques based on hyperplane bisections. These are used, for instance, in the proof of Gromov's waist inequality.

In the second part of the course we will study L^1-optimal transport in Riemannian manifolds, with emphasis on the role of curvature. This method is used, for instance. in the proof of the isoperimetric inequality for Riemannian manifolds whose Ricci curvature is bounded from below.

## Learning Outcomes

The students will study the basic notions and theorems in Optimal Transport and Convex Localization. This will allow them to apply these methods in their research, and to access advanced literature on the subject.