Instructor: Jan Maas
Teaching Assistant: Dominik Forkert
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.
A solid background in analysis will be helpful.
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 & 18, March 23, April 1, April 13 & 15, April 20
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 |