math566:Theory of Optimal Transportation

(Term 2, 2013/2014: Jan, 2014 -- April, 2014)

MATH 566:201 Theory of Optimal Transportation
(UBC registra course page here.)

Class: MWF 14:00--15:00 pm.
Location: MATH 103

Office hours (subject to change): TBA. please email at yhkim "at" math "dot" ubc 'dot' ca

Assignments

Announcements

Some papers you can choose to present:
[Some fundamental papers are skipped and replaced with follow-up works if the latter are easier to read.]
You should discuss with me ahead about your choice of a paper.

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* Existence of OT maps.

- [Kantorovich]. The first paper in the handout.

- [McCann]. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309-323, (1995).

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* Regularity of OT.

[A.D. Alexandrov] 2d regularity result. The last paper in the handout.

For more ambitious students, especially if you are not familiar with regularity theory of fully nonlinear equations. :

- [Figalli and DePhilippis]. W^{2,1} regularity for solutions of the Monge-Ampère equation. Invent. Math., 192 (2013), no. 1, 55-69.

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* Regularity theory of optimal transport for general cost functions

[Ma, Trudinger and Wang] Regularity of potential functions of the optimal transportation problem Archive for rational mechanics and Analysis 2005

[Loeper] On the regularity of solutions of optimal transportation problems Acta mathematica, 2009

[Kim and McCann] Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. (JEMS) 12 (2010) 1009-1040.

[Kim] Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn120, 15 pages, doi:10.1093/imrn/rnn120.

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* Displacement interpolation/convexity, gradient flows, and functional inequalities.

[McCann] A convexity principle for interacting gases. Adv. Math. 128, 153-179 (1997)
-- This is the paper where displacement convexity is found.

[Otto and Villani] Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality Journal of Functional Analysis, Volume 173, Issue 2, 1 June 2000, Pages 361–400.

[Sturm and von Renesse] Transport Inequalities, Gradient Estimates, Entropy and Ricci Curvature, with Max-Konstantin von Renesse. In Comm. Pure Appl. Math. 68,923-940 2005.

A harder but rewarding one is
[McCann, Cordero-Erausquin and Schmuckenschlaeger]
A Riemannian interpolation inequality a la Borell, Brascamp and Lieb. Invent. Math. 146 (2001) 219-257

Long, but, very interesting paper:
[Sturm] The Space of spaces: curvature bounds and gradient flows on the space of metric measure spaces . Preprint.

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* Isoperimetric inequalitites:

[McCann]
Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys. 195, 699-723 (1998)
--This is a pioneering work.

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* Discrete Math and Optimal Transport:

[Villani. and Ollivier]: A curved Brunn–Minkowski inequality on the discrete cube — or: What is the Ricci curvature of the discrete hypercube? To appear in SIAM J. on Discrete Math. Download

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* Polar factorization:

[McCann] Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608
-- This extends Brenier's results to Riemannian manifolds.

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* Volume preserving maps:
[Brenier] Geodesics on groups of volume preserving maps (pdf file-See final version in CPAM 1999)

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* Multimarginal Optimal Transport problems.

[Kim and Pass] Multi-marginal optimal transport on Riemannian manifolds.
- Good for students with geometry background

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* Partial transport and free boundary:

[Figalli] The optimal partial transport problem, Arch. Ration. Mech. Anal., 195 (2010), no. 2, 533-560.

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* Relativistic point of view:

[Bertrand and Puel] The optimal mass transport problem for relativistic costs. Calculus of Variations and Partial Differential Equations January 2013, Volume 46, Issue 1-2, pp 353-374

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* Gradient flows in physical phenomena:

[Robert V. Kohn and Felix Otto] Upper bounds on coarsening rates. Comm. Math. Phys., 229(3):375-395, 2002.
--For some details about coarsening, see http://www.mis.mpg.de/applan/research/coarsening.html

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* Density Functional Theory of electronic structure of atoms and molecules:

[C.Cotar, G.Friesecke, C.Klüppelberg] Density functional theory and optimal transportation with Coulomb cost. CPAM, to appear. http://arxiv.org/abs/1104.0603
* Additional references: since Feb. 5:

[Figalli, Maggi, Pratelli] A refined Brunn–Minkowski inequality for convex sets Ann. I. H. Poincaré – AN 26 (2009) 2511–2519

[DePhilippis-Figalli] Optimal regularity of the convex envelop , TAMS, to appear.
: a very nice result, and seems very well-written.

[Klartag, Kolesnikov] Eigenvalue distribution of optimal transportation. arXiv:1402.2636.
: seems very interesting!

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Schedule / Plan / Progress / Summary (Subject to change)

Week	Date	Contents (Sections refer to [Villani, Topics in optimal transportation] and reference are from the handout. e.g. [Kantorovich]= the paper by Kantorovich.
1	Jan. 6.	Basic set up of the optimal transport problem. c-monotonicity.
	Jan. 8.	c-cyclical monotonicity. convex functions.
	Jan 10.	Rockafellar's theorem.
2	Jan 13.	Kantorovich problem. Existence of optimal transport plan. c-cyclical monotonicity of optimal transference plan, 2.4.1--2.4.4. [McCann, Existence and uniqueness of monotone measure-preserving maps.] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)]
	Jan 15.	c-cyclical monotonicity of optimal transference plan, 2.4.1--2.4.4. c-convex functions, c-subdifferential. 2.5 Existence and uniqueness in Monge-Kantorovich problem. [McCann, Existence and uniqueness of monotone measure-preserving maps. ] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)]
	Jan 17	Existence and uniqueness in Monge-Kantorovich problem. [McCann, Existence and uniqueness of monotone measure-preserving maps. ] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)]
3	Jan 20	Monge-Ampere equations [Villani, Ch. 4]
	Jan 22	Monge-Ampere equations [Villani, Ch. 4] Isoperimetric inequality [Villani Ch. 6]
	Jan 24	Second differentiability of convex functions (Alexandrov theorem).
4	Jan 27	Log-Sobolev inequality [Villani Ch. 6]
	Jan 29	Brunn-Minkowski inequality
	Jan 31	Prekopa-Leindler inequality [Villani Ch. 6]
5	Feb 3	Geometry of the space of probability measures: Wasserstein metric [Villani, Ch 7]
	Feb 5	Geometry of the space of probability measures: displacement interpolation. [Villani, Ch 7]
	Feb 7	Geometry of the space of probability measures: displacement interpolation and displacement convexity. Interacting gases. [Villani, Ch 7]
6	Feb 10	No class: Family day
	Feb 12	Displacement convexity of functionals on the space of probability measures. Entropy
	Feb 14	Applications of displacement convexity.
7	Feb 17	Midterm break
	Feb 19	Midterm break
	Feb 21	Midterm break
8	Feb 24
	Feb 26
	Feb 28
9	Mar 3
	Mar 5
	Mar 7
10	Mar 10
	Mar 12
	Mar 14
11	Mar 17
	Mar 19
	Mar 21
12	Mar 24
	Mar 26
	Mar 28
13	Mar 31
	Apr 2
	Apr 4
14	Apr 7	last class