Optimal Transport and applications

Period of duration of course
Course info
Number of course hours
Number of hours of lecturers of reference
Number of hours of supplementary teaching

Type of exam

Oral exam and seminars


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PhD students or undergraduate (master level)


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

Educational aims

The course originates from the teaching experience of Luigi Ambrosio in the Scuola Normale Superiore, where it has been given many times during the last 20 years. The topics and the tools have been chosen at a sufficiently general and advanced level, so that the student or scholar interested to a more specific theme will gain from the book the necessary background to explore it. After a large and detailed introduction to the classical theory a more specific attention is devoted to applications to Geometric and Functional inequalities and to Partial Differential Equations.

Bibliographical references

L.Ambrosio, E.Brue', D.Semola  Lectures on Optimal Transport, UNITEXT, 2022