General Prerequisites: Good command of Part A Integration, Probability and Differential Equations 1 are essential; the main concepts which will be used are the convergence theorems and the theorems of Fubini and Tonelli, and the notions of measurable functions, integrable functions, null sets and L\(^p\) spaces. The Cauchy-Lipschitz theory and Picard's theorem proofs will be used. Basic knowledge of random variables, laws, expectations, and independence are needed. A good working knowledge of Part A Core Analysis (metric spaces) is expected. Knowledge of B8.1 Probability, Measure and Martingales will certainly help but it is not essential.

Course Overview: This course will serve as an introduction to optimal transportation theory, its application in the analysis of PDE, and its connections to the macroscopic description of interacting particle systems.

Learning Outcomes: Getting familar with the Monge-Kantorovich problem and transport distances. Derivation of macroscopic models via the mean-field limit and their analysis based on contractivity of transport distances. Dynamic Interpretation and Geodesic convexity. A brief introduction to gradient flows and examples.

Course Synopsis: 1. Interacting Particle Systems & PDE:

- Granular Flow Models and McKean-Vlasov Equations.

- Nonlinear Diffusion and Aggregation-Diffusion Equations.

2. Optimal Transportation: The metric side

- Functional Analysis tools: weak convergence of measures. Prokhorovâ€™s Theorem. Direct Method of Calculus of Variations.

- Monge Problem. Kantorovich Duality.

- Transport distances between measures: properties. The real line. Probabilistic Interpretation: couplings.

3. Mean Field Limit & Couplings

- Continuity Equation: measures sliding down a convex valley.

- Dobrushin approach: derivation of the Aggregation Equation.

- Boltzmann Equation for Maxwellian molecules: Tanaka Theorem.

4. An Introduction to Gradient Flows

- Dynamic Interpretation of optimal tranport.

- McCannâ€™s Displacement Convexity: Internal, Interaction and Confinement Energies.

- Gradient Flows: Differential and metric viewpoints.