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Optimal Transportation

Instructor: Jan Maas

Teaching Assistant: Dominik Forkert

Description

 

The Monge-Kantorovich problem of optimal transport is to transfer mass from a given initial distribution to a prescribed target distribution in such a way that the total transport costs are minimized. This problem has originally been motivated by applications in engineering and economics. In the last decades, several unexpected connections have been discovered between optimal transport and seemingly unrelated problems in analysis, probability theory and geometry. 

This course presents an overview of these developments. Some of the topics that we will discuss are

* Gradient flow methods for evolution equations

* Ricci curvature bounds for metric measure spaces

* Functional inequalities, isoperimetry, and concentration of measure

We shall cover parts of the following books:

* [TOT] Cédric Villani, Topics in Optimal Transportation, AMS 2003

* [OTON] Cédric Villani, Optimal Transport: Old and New, Springer 2009

* [UGOT] Luigi Ambrosio and Nicola Gigli, A User's Guide to Optimal Transport

Lectures take place on Mondays and Wednesdays, 8:45-10:00 in Mondi 2, starting on 9 March 2015.

 

 
Requirements

A solid background in analysis will be helpful.

 

 

Brief contents of the lectures

March 9 

References: [OTON] Chapter 3, "The founding fathers of optimal transport"; [TOT] Introduction (p. 1-9)

March 11 

References: A nice reference for weak convergence of probability measures on separable metric spaces is Chapter 11 in the classical book by Dudley: Real Analysis and Probabilty; see also the detailed lecture notes by van Gaans. The existence of a minimizer for the Kantorovich problem is treated in [UGOT] Section 1.1.

 March 16  

References: [UGOT] Section 1.2

 March 18

References: [UGOT] Sections 1.2 and 1.3

March 23

References: [UGOT] Sections 1.2 and 1.3; [TOT] Sections 1.2 and 1.4

April 1

References: [TOT] Section 2.1, p. 66-70

April 13  

References: [TOT] Chapter 3

April 15 

References: [TOT] Chapter 3, [UGOT] Section 2.1

April 20

References: [UGOT] Section 2.1

April 22

References: [UGOT] Sections 2.1 and 2.2

April 27

References: [UGOT] Section 2.2

April 29

References: [UGOT] Section 2.3

May 4

References: [UGOT] Section 2.3

May 11

References: [TOT] Sections 5.1 and 5.2

May 13 

References: [TOT] Sections 5.2 and 6.1

May 18

References: [TOT] Sections 6.3

May 20

References: C. Cotar, G. Friesecke and C. Klüppelberg; Comm. Pure Appl. Math. 66 (2013), no. 4, 548–599.

June 1

References:
D. Cordero-Erausquin: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161 (2002), no. 3, 257–269.
D. Cordero-Erausquin, B. Nazaret, C. Villani: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182 (2004), no. 2, 307–332. 

June 3

References: F. Otto: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001), no. 1-2, 101–174.

June 15

References: [UGOT] Section 4

June 17

References: [UGOT] Section 4

July 6

 References: [UGOT] Section 4

Notes (by Marcin Napiórkowski): March 9 & 11, March 16 & 18March 23, April 1, April 13 & 15, April 20 

Exercises

  Deadline
Exercise sheet 1 March 23, 2015
Exercise sheet 2 April 13, 2015
Exercise sheet 3 April 27, 2015
Exercise sheet 4 May 11, 2015
Exercise sheet 5 June 1, 2015
Exercise sheet 6 June 15, 2015
Exercise sheet 7 June 29, 2015