In the talk, I will first present a typical Mean Field Game problem, as in the theory introduced by Lasry-Lions and Huang-Caines-Malhamé, concentrating on the case where the game has a variational structure (i.e., the equilibrium can be found by minimizing a global energy) and is purely deterministic (no diffusion, no stochastic control). From the game-theoretical point of view, we look for a Nash equilibrium for a non-atomic congestion game, involving a penalization on the density of the players at each point. I will explain why regularity questions are natural and useful for rigorously proving that minimizers are equilibria, making the connection with what has been done for the incompressible Euler equation in the Brenier's variational formalism. I will also introduce a variant where the penalization on the density is replaced by a constraint, which lets a price (which is a pressure, in the incompressible fluid language) appears on saturated regions. Then, I will sketch some regularity results which apply to these settings.
The content of the talk mainly comes from joint works with A. Mészáros, P. Cardaliaguet, and H. Lavenant.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

Optimal transport, a mathematical theory which developed out of a problem raised by Gaspard Monge in the 18th century and of the reformulation that Leonid Kantorovich gave of it in the 20th century in connection with linear programming, is now a very lively branch of mathematics at the intersection of analysis, PDEs, probability, optimization and many applications, ranging from fluid mechanics to economics, from differential geometry to data sciences. In this short course we will have a very basic introduction to this field. The first lecture (2h) will be mainly devoted to the problem itself: given two distributions of mass, find the optimal displacement transforming the first one into the second (studying existence of such an optimal solution and its main properties). The second one (2h) will be devoted to the distance on mass distributions (probability measures) induced by the optimal cost, looking at topological questions (which is the induced topology?) as well as metric ones (which curves of measures are Lipschitz continuous for such a distance? what can we say about their speed, and about geodesic curves?) in connection with very natural PDEs such as the continuity equation deriving from mass conservation.[-]

Spin models are lattice models that describe magnetic properties of materials. In this talk we will examine a 2-dimensional planar spin model (known as the J1-J2-J3 model) which exhibits frustration. Frustration is the phenomenon due to conflicting interatomic ferromagnetic/antiferromagnetic interactions that prevent the energy of every pair of interacting spins to be simultaneously minimized. The frustration mechanism leads to complex geometric patterns in the material. We study these complex geometric patterns by carrying out a discrete-to-continuum variational analysis as the lattice spacing tends to zero, finding the energetic regime for which many chiralphases can coexist. In particular, we will show that the surface tension between the chiral phases is captured by a continuum energy obtained by suitably selecting solutions to the eikonal equation. The results presented in the seminar are based on works in collaboration with M. Cicalese and M. Forster.[-]

This talk revolves around two variational models in finite crystal plasticity where the possible deformations are restricted to plastic glide along one active slip system. In the first model for polycrystals, our focus lies on the set of attainable macroscopic strains, whose analysis is linked to the solvability of an inhomogeneous differential inclusion problem with affine boundary values. We discuss how to estimate this set by exploiting admissible boundary interaction, global compatibility, and the interplay between the slip mechanism and the polycrystalline texture. The second model describes high-contrast composites with periodically arranged layers and gives rise to energy functionals with non-convex differential constraints. We prove homogenization theorems via $\Gamma$-convergence in the Sobolev and BV setting and study the resulting limit models, addressing the uniqueness of minimizers and deriving necessary conditions. These results are joint work with Fabian Christowiak (University of Regensburg), Elisa Davoli (TU Vienna), Dominik Engl (KU Eichstätt-Ingolstadt), and Rita Ferreira (KAUST).[-]