In this talk, I will discuss my joint paper with Olivier Biquard and Paul Gauduchon on ALF Ricci-flat Riemannian  4-manifolds. My collaborators had previously classified all such spaces that are toric and Hermitian, but not Kaehler. Our main result uses an open curvature condition to prove a rigidity result of the following type: any Ricci-flat metric that is sufficiently close to a non-Kaehler, toric, Hermitian ALF solution (with respect to a norm that imposes reasonable fall-off at infinity) is actually  one of the  known Hermitian toric  solutions. 
 

Studying the topology of zero sets of maps is a central topic in many areas of mathematics. Classical homological invariants, such as Betti numbers, are not always suitable for this purpose due to the fact that they do not distinguish between topological features of different sizes. Topological data analysis provides a way to study topology coarsely by ignoring small-scale features. This approach yields generalizations of a number of classical theorems, such as Bézout's theorem and Courant’s nodal domain theorem, to a wider class of maps. We will explain this circle of ideas and discuss potential directions for future research. The talk is partially based on joint works with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

Using novel arguments as well as techniques developed over the last  twenty years to study mean field games, in this paper (i) we investigate the optimal control of the Dyson equation, which is the mean field equation for the so-called Dyson Brownian motion, that is, the stochastic particle system satisfied by the eigenvalues of large random matrices, (ii) we establish the well-posedness of the resulting infinite dimensional Hamilton-Jacobi equation, 
(iii) we provide a complete and direct proof for the large deviations for the spectrum of large random matrices, and (iv) we study the asymptotic behavior of the transition probabilities of the Dyson Brownian motion.  Joint work with Charles Bertucci and Pierre-Louis Lions.