MATH
566:101
Theory
of Optimal Transportation
(UBC registra course page here.)
Class:
M W 3:00--4:30 pm.
Location: MATX 1118
Office
hours (subject to change): MWF 11 -- 12 or by
appointment: please email at yhkim "at" math "dot" ubc 'dot'
ca
Assignments
HW 2 is now due on Monday, Oct. 31.
HW 1 is now due on Friday, Oct. 7.
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 | ||
Organization meeting |
||
2 | Mon, Sept 12 |
Basic set-up: Monge problem. c-monotonicity. c-cyclical
monotonicity. Reading: Introduction. 2.2, 2.3 |
Wed, Sept 14 |
cyclical monotonicity, 2.3. subdifferential of semi-convex function, 2.1.3 Rockafellar's theorem, 2.3.2 see also [Rockafellar]. Introduction to Kantorovich problem: transference plans. [Kantorovich] |
|
3 | Mon, Sept 19 |
Existence of optimal transference plan, c-cyclical monotonicity of optimal transference plan, 2.4.1--2.4.4 [McCann] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)] |
Wed, Sept 21 |
c-convex functions, c-subdifferential. 2.5 Existence and uniqueness in Monge-Kantorovich problem. [McCann] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)] |
|
4 | Mon, Sept 26 |
Existence and uniqueness in Monge-Kantorovich problem. [McCann] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)] |
Wed, Sept 28 |
Existence and uniqueness in Monge-Kantorovich problem. Brenier's theorem. [McCann] [Gangbo-McCann, The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)] |
|
5 | Mon, Oct 3 |
Polar factorization [Villani, Ch. 3] Monge-Ampere equations [Villani, Ch. 4] |
Wed, Oct 5 |
Monge-Ampere equations [Villani, Ch. 4] Isoperimetric inequality [Villani Ch. 6] |
|
Fri, Oct 7 |
HW
1 is Due. Place it in my mailbox |
|
6 | Mon, Oct 10. No class |
Thanksgiving day |
Oct 12 (Wed) | Isoperimetric inequality: Log-Sobolev inequality [Villani Ch. 6] |
|
7 |
Oct 17 (Mon) |
Brunn-Minkowski inequality Prekopa-Leindler inequality [Villani Ch. 6] |
Oct 19 (Wed) | Existence of optimal transport plann on R^n. |
|
8 |
Oct 24 (Mon) |
Geometry of the space of probability measures: Wasserstein metric [Villani, Ch 7] |
Oct 26 (Wed) | Geometry of the space of probability measures: Wasserstein metric [Villani, Ch 7] |
|
9 |
Oct 31 (Mon) |
Displacement interpolation and Displacement convexity [Villani, Ch. 5] |
Nov 2 (Tue) |
Displacement interpolation and Displacement convexity [Villani, Ch. 5] | |
10 | Nov 7 (Mon) |
no class: Displacement interpolation and Displacement convexity [Villani, Ch. 5] |
Nov 9 (Wed) | Displacement interpolation and Displacement convexity [Villani, Ch. 5] |
|
11 | Nov 14 (Mon) | Plan: Differential geometry of the space of probability measures [Villlani, Ch. 8] |
Nov 16 (Wed) | Plan: Otto calculus and gradient flows [Villani, 8.1.2, 8.2, 8.3] | |
12 | Nov 21(Mon) | Plan: tiem-discretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the Fokker-Plank Equation, SIAM J. of Math. Anal. 29 (1998), 1-17.] |
Nov 23 (Wed) | Plan: tiem-discretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the Fokker-Plank Equation, SIAM J. of Math. Anal. 29 (1998), 1-17.] |
|
13 | Nov 28 (Mon) | Plan: tiem-discretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the Fokker-Plank Equation, SIAM J. of Math. Anal. 29 (1998), 1-17.] |
Nov 30 (Wed) | Plan: Second derivatives in Wasserstein geometry, Log-Sobolev and transport inequalities [Villani, 8.2, 8.3, 9.1, 9.2, 9.3, 9.4] | |