# Winter school

There will be a two-week winter school for PhD students and postdocs, on February 11-22. The goal is to give introductions to current research areas involving calculus of variations. There will be 3 courses per week, as well as some participant talks.

The speakers will be:

** Week 1** (Feb. 11-15): Dorin Bucur (U. Savoie), Félix Otto (MPIM Leipzig) and Vlad Vicol (Princeton).

__ Week 2__ (Feb. 18-22): Luigi Ambrosio (SNS Pisa), François Delarue (U. Nice) and Jan Maas (IST Vienna).

To register as a participant, please send an email to the organizers at trimestre-CIMI-2019[at]math[dot]univ-toulouse[dot]fr . While the school will be adressed to junior researchers, more senior participants are also welcome.

Financial support is available for junior participants, to cover expenses for local accommodation. If you wish to apply, please indicate it in the email to the organizers, and send a CV and a cover letter. No financial support for the cost of traveling to and from Toulouse is available at this time. Priority will be given to participants staying for both weeks. If you wish to give a participant talk, please tell us, but note that the number of slots will be limited. The deadline to apply for funding is November 15th.

Talks will start on Monday mornings, and end on Fridays in the early afternoon.

__Titles and abstracts: __

__Luigi Ambrosio:__ Asymptotic behaviour of the matching problem via PDE techniques.

In the lectures I will present recent work on the asymptotic analysis of the matching problems (both in the semi-discrete and bipartite case) in the challenging case of two space dimensions. The PDE strategy is based on the construction of nearly optimal matchings by linearizingthe Monge-Ampere equation associated to a short-time regularization of the empirical measures. A refined analysis is involved in the analysis of the errors due to the linearization and to the short-time regularization.

__Dorin Bucur:__ A free boundary approach to spectral shape optimization problems

In these lectures, spectral inequalities of isoperimetric type will be seen from a shape optimisation point of view. The main examples are the minimization of an eigenvalue of the Dirichlet (or Robin) Laplacian under a volume constraint. I will present the classical approach in the Gamma convergence framework to prove existence of solutions, and recent developments based on free boundary/discontinuity techniques to extract qualitative information on the optimal shapes. Precisely, I will develop the notions of shape sub and supersolutions and show how the analysis of a general shape optimization problem of spectral type can be reduced to particular free boundary problems. In particular, for the Robin boundary conditions, I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal set.

__François Delarue__: Mean field games. Convergence problem and regularisation.

The lectures will be dedicated to mean field games theory, the aim of which is to address Nash equilibria within large populations of players interacting with one another in a mean field way. The theory was introduced independently by Lasry and Lions and by Huang, Caines and Malhamé in 2006. Here is the plan:

1) I will present the concept, together with the connection with mean field optimal control (and optimal transportation).

2) I will give basic existence results, together with some cases where uniqueness is known to hold.

3) I will focus on the proof of the convergence problem: Proving that equilibria to the finite player game do converge to the asymptotic mean field formulation (as the number of players tends to infinity).

4) I will address regularisation methods to restore uniqueness of the equilibria.

Bibliography:

P. Cardaliaguet. Notes from P.L. Lions' lectures at the College de France. Technical report, https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf, 2012.

P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions. The master equation and the convergence problem in mean field games. To appear in Annals Maths Studies.

R. Carmona and F. Delarue. Probabilistic Theory of Mean Field Games: Vol. I, Mean Field FBSDEs, Control, and Games. Stochastic Analysis and Applications. Springer Verlag, 2018.

R. Carmona and F. Delarue. Probabilistic Theory of Mean Field Games: Vol. II, Mean Field Games with Common Noise and Master Equations. Stochastic Analysis and Applications. Springer Verlag, 2018.

F. Delarue. Restoring uniqueness to mean field games by randomizing the equilibria. Technical report, https://arxiv.org/abs/1804.03561, 2018.

M. Huang, P.E. Caines, and R.P. Malhame. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information and Systems, 6:221-252, 2006.

J.M. Lasry and P.L. Lions. Jeux a champ moyen i. le cas stationnaire. Comptes Rendus de l'Academie des Sciences de Paris, ser. A, 343(9), 2006.

P.L. Lions. Theorie des jeux a champs moyen et applications. Lectures at the College de France. http://www.college-de-france.fr/default/EN/all/equder/cours et seminaires.htm, 2007-2008.

__Jan Maas: __Recent advances in dynamical optimal transport

Optimal transport continues to be a very active field of research at the interface of analysis, probability and geometry. In this lecture series we present an overview of dynamical optimal transport, and some of its applications to discrete probability, chemical reaction networks, and non-commutative analysis. Particular focus is on recent results on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation of dynamical optimal transport.

__Felix Otto:__ Stochastic homegenization

__Vlad Vicol:__ Convex integration and applications in incompressible hydrodynamics

We consider applications of the method of convex integration, as pioneered by De Lellis-Szekelyhidi, to the Euler and Navier-Stokes equations. Topics discussed will include the Onsager conjecture and the nonuniqueness of finite energy distributional solutions to the Navier-Stokes equations.