Young-Heon Kim’s  Teaching              Home           Research

(Term 1, 2011/2012: September, 2011 -- December, 2011)

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

Syllabus


Assignments

HW 2 is now due on Monday, Oct. 31.

HW 1 is now due on Friday, Oct. 7.

Announcements

You can use the online survey form to give me an annonymous comments/questions, etc. 
The link is http://www.surveymonkey.com/s/YSC29YT    You need the password that I have sent to you in a separate email.


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


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]