Optimal Transport and Applications

Registro delle lezioni

Anno accademico 2023/2024
Docente Luigi Ambrosio

Lecture

  • 03 Nov 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Introduction to the optimal transport problem. Basic notation and measure-theoretic notions

  • 17 Nov 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Weak and strong topologies and transport maps. Disintegration theorem, structure of transport plans. Kantorovich's formulation of the optimal transport problem

  • 24 Nov 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Weak convergence and Prokohorov theorem. Existence of optimal plans. Convex duality

  • 01 Dic 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Classical duality and c-duality, proof of duality via Fenchel-Rockafellar, c-monotonicity as necessary optimality condition

  • 07 Dic 2023 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Necessary and sufficient optimality conditions in Optimal Transport

  • 12 Gen 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Existence of optimal maps in the case of the cost equal to the p-th power of the Euclidean distance

  • 19 Gen 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Polar factorization of vector fields, Knothe map, formal proof of the isoperimetric inequality, fine properties of convex functions

  • 26 Gen 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Rigorous proof of the isoperimetric inequality, Caffarelli's example, stability of optimal plans, convergence in Probability, stability of optimal maps

  • 02 Feb 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Monge-Ampere measure and regularity results. Optimal Transport on Riemannian Manifolds

  • 09 Feb 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    The metric side of Optimal Transport, convergence and completeness

  • 16 Feb 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Absolutely continuous functions, Metric derivative, action, length. Dynamic formulation of the optimal transport problem.

  • 01 Mar 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Lower semicontinuity of the action, basic compactness criteria and lifting. Geodesics in the space of probabilities versus probabilities in the space of geodesics

  • 08 Mar 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - In presenza

    Positive and negative curvature in the sense of Alexandrov. Nonbranching metric spaces. Gradient flows, basic definition and lambda-convexity

  • 15 Mar 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    EVI formulation of gradient flows and equiivalence. Minimal selection and slope. Basic properties of gradient flows.

  • 22 Mar 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Basic regularization properties of gradient flows. Discretization and interpolation schemes, discussion about discrete version of the Evolution Variational Inequality

  • 05 Apr 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Cauchy type estimates of gradient flows and existence of solutions. Divergence and gradient in differentiable manifolds. Examples of gradient flows: heat equation in bounded domains, p-laplace equation and Fokker-Planck equation

  • 12 Apr 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Realization of PDE's as gradient flows, Negative Sobolev spaces. EDE and EDI solutions of gradient flows in metric spaces

  • 19 Apr 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    EDI and EDE gradient flows. Upper gradients. Existence of EDI solutions

  • 03 Maggio 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Convexity and lower semicontinuity of integral functionals defined on the space of probability measures

  • 10 Maggio 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Proof of functional inequalities via Optimal Transport. Theory of characteristics for continuity and transport equations

  • 17 Maggio 2024 (2h 00m)

    Luigi Ambrosio - Corso (attività didattica) - Mista

    Hopf-Lax semigroup, differentiable and Riemannian structure of the Wasserstein space, applications to gradient flows