MATH
566:101
Theory
of Optimal Transportation
(UBC registra course page here.)
Class:
M W 3:004:30 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 setup: Monge problem. cmonotonicity. ccyclical
monotonicity. Reading: Introduction. 2.2, 2.3 
Wed, Sept 14 
cyclical monotonicity, 2.3. subdifferential of semiconvex 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, ccyclical monotonicity of optimal transference plan, 2.4.12.4.4 [McCann] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 
Wed, Sept 21 
cconvex functions, csubdifferential. 2.5 Existence and uniqueness in MongeKantorovich problem. [McCann] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 

4  Mon, Sept 26 
Existence and uniqueness in MongeKantorovich problem. [McCann] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 
Wed, Sept 28 
Existence and uniqueness in MongeKantorovich problem. Brenier's theorem. [McCann] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 

5  Mon, Oct 3 
Polar factorization [Villani, Ch. 3] MongeAmpere equations [Villani, Ch. 4] 
Wed, Oct 5 
MongeAmpere 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: LogSobolev inequality [Villani Ch. 6] 

7 
Oct 17 (Mon) 
BrunnMinkowski inequality PrekopaLeindler 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: tiemdiscretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the FokkerPlank Equation, SIAM J. of Math. Anal. 29 (1998), 117.] 
Nov 23 (Wed)  Plan: tiemdiscretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the FokkerPlank Equation, SIAM J. of Math. Anal. 29 (1998), 117.] 

13  Nov 28 (Mon)  Plan: tiemdiscretization scheme for gradient flow
[Villani, 8.4] [Jordan, Kinderlehrer, Otto, The Variational Formulation of the FokkerPlank Equation, SIAM J. of Math. Anal. 29 (1998), 117.] 
Nov 30 (Wed)  Plan: Second derivatives in Wasserstein geometry, LogSobolev and transport inequalities [Villani, 8.2, 8.3, 9.1, 9.2, 9.3, 9.4]  