MATH
566:201
Theory
of Optimal Transportation
(UBC registra course page here.)
Class:
MWF 14:0015:00 pm.
Location: MATH 103
Office
hours (subject to change): TBA. please email at
yhkim "at" math "dot" ubc 'dot' ca
Assignments
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.
cmonotonicity. 
Jan. 8. 
ccyclical monotonicity. convex functions. 

Jan 10. 
Rockafellar's theorem.  
2  Jan 13. 
Kantorovich problem. Existence of optimal transport plan. ccyclical monotonicity of optimal transference plan, 2.4.12.4.4. [McCann, Existence and uniqueness of monotone measurepreserving maps.] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 
Jan 15. 
ccyclical monotonicity of optimal transference plan,
2.4.12.4.4. cconvex functions, csubdifferential. 2.5 Existence and uniqueness in MongeKantorovich problem. [McCann, Existence and uniqueness of monotone measurepreserving maps. ] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 

Jan 17 
Existence and uniqueness in MongeKantorovich problem. [McCann, Existence and uniqueness of monotone measurepreserving maps. ] [GangboMcCann, The geometry of optimal transportation. Acta Math. 177, 113161 (1996)] 

3  Jan 20 
MongeAmpere equations [Villani, Ch. 4] 
Jan 22 
MongeAmpere equations [Villani, Ch. 4] Isoperimetric inequality [Villani Ch. 6] 

Jan 24 
Second differentiability of convex functions (Alexandrov
theorem). 

4  Jan 27 
LogSobolev inequality [Villani Ch. 6] 
Jan 29 
BrunnMinkowski inequality  
Jan 31 
PrekopaLeindler 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 