Optimal Transport and Applications

Lecture I

1.1 Notation and preliminary results 

1.2. Monge’s formulation of the optimal transport problem 

Lecture II 

2.1 Kantorovich’s formulation of the optimal transport problem

2.2 Transport plans versus transport maps

2.3 Advantage so Kantorovich’sformulation

2.4 Existence of optimalplans

Lecture III 

3.1 Convex Analysis tools 

3.2 Proof of duality via Fenchel-Rockafellar

3.3 The theory of c-duality

3.4 Proof of duality and dual attainment for bounded and continuous cost functions

Lecture IV 

4.1 Duality and necessary/sufficient optimality conditions for lower semicontinuous costs

4.2 Remarks about necessary and sufficient optimality conditions 

4.3 Remarks about c-cyclical monotonicity, c-concavity and c-transforms

4.3.1 Cost=distance^2 

4.3.2 Cost=distance 

4.3.3 Convex costs on the real line 

Lecture V 

5.1 Existence of optimal transport maps 

5.2 A digression about Monge’s problem 

5.3 Applications

5.4 Iterated monotone rearrangement

Lecture VI 

6.1. Isoperimetric inequality 

6.2 Stability of optimal plans and maps 

7 Lecture VII 

7.1 A general change of variables formula 

7.2 The Monge–Ampère equation 

7.3 Optimal transport on Riemannian manifolds. 

8 Lecture VIII 

8.1 The distance W_2 inP_2(X) 

8.2 Completeness of (P2(X),W2)

8.3 Characterization of convergence in (P_2(X),W2) and applications

9 Lecture IX

9.1 Absolutely continuous curves and their metric derivative 

9.2 Geodesics and action

9.3 Dynamic reformulation of the optimal transport problem 

10 Lecture 

10.1 Lower semicontinuity of the action 

10.2 Compactness criterion for curves and random curves

10.3 Lifting of geodesics from X to P_2(X) 

11 Lecture XI 

11.1 lambda-convex functions 

11.2 Differentiability of absolutely continuous curves 

11.3 Gradient flows 

12 Lecture XII 

12.1 Maximal monotone operators 

12.2 The implicit Euler scheme 

12.3 Reduction to initial conditions with finite energy 

12.4 Discrete EVI

13 Lecture XIII 

13.1 p-Laplace equation, heat equation in domains, Fokker-Planck equation 

13.2 The heat equation in Riemannian manifolds

13.3 Dual Sobolev space and heat flow

14 Lecture XIV 

14.1 EDE, EDI solutions and upper gradients 

4.2 Existence of EDE, EDI solutions

14.3 Proof of Theorem 14.7 via variational interpolation 

15 Lecture XV 

15.1 Semicontinuity of internal energies 

15.2 Convexity of internal energies

15.3 Potential energy functional

15.4 Interaction energy 

15.5 Functional inequalities via optimal transport 

16 Lecture XVI 

16.1 Continuity equation and transport equation

16.2 Continuity equation of geodesics in the Wasserstein space 

16.3 Hopf-Lax semigroup 

17 Lecture XVII 

17.1 Benamou-Brenier formula 

17.2 Correspondence between absolutely continuous curves in P_2(R^n) and solutions to the continuity equation

18 Lecture XVIII 

18.1 Otto’s calculus

18.2 Formal interpretation of some evolution equations as Wasserstein gradient flows

18.3 Rigorous interpretation of the heat equation as a Wasserstein gradient flow 

18.4 Morerecent ideas and developments

19 Lecture XIX 

9.1 Heat flow on Riemannian manifolds 

19.2 Heat flow, Optimal Transport and Ricci curvature