Random regular graphs form a ubiquitous model for chaotic systems. However, the spectral properties of their adjacency matrices have proven difficult to analyze because of the strong dependence between different entries. In this talk, I will describe recent work that shows that despite this, the fluctuation of eigenvalues of the adjacency matrix are of the same order as for Gaussian matrices. This gives an optimal error term for Friedman's theorem that the second eigenvalue of the adjacency matrix of a random regular graph converges to the spectral radius of an infinite regular tree. Crucial is tight analysis of the Greenâ??s function of the adjacency operator and an analysis of the change of the Green's function after a random edge switch. This is based on joint work with Jiaoyang Huang and Horng-Tzer Yau.

The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing controlled multi-agent systems. The study of such systems has seen increased interest in recent years, due to their ubiquity in applications coming from macroeconomics, social behavior, and telecommunications. When the number of agents becomes large, the model can be formally replaced by one involving a mean-field description of the population, analogously to similar models in statistical physics. Justifying this continuum limit is often nontrivial and is sensitive to the type of stochastic noise influencing the population, i.e. idiosyncratic or systemic. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces. In particular, we develop new stability and regularity results for the equations. These allow for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type. We conclude with a discussion of some further problems for which the techniques for equations on Wasserstein space may be amenable.