In this talk, we will focus on a conjecture due to Naor concerning an isomorphic reverse isoperimetry phenomenon, and explore its close connection to random matrix theory. Namely, we will see that the resolution of both the weak and strong versions of the conjecture in the setting of unitary ideals require sharp non-asymptotic estimates for various moments associated with log-concave unitarily invariant random matrices, which may also be of independent interest. Based on joint works with Naor, and with Naor and Paouris.

In 2004, motivated by connections of random matrix theory to number theory, Diaconis and Gamburd showed a fascinating connection between the enumeration problem of magic squares (squares filled with integers with row and column sum constraints) and the moments of the ‘secular coefficients’ of random matrices, when the size of the matrix tends to infinity. These are the coefficients in the monomial expansion of a characteristic polynomial, or equivalently, the elementary symmetric polynomials of the eigenvalues of this random matrix. It turns out that this characteristic polynomial has a limit, when the matrix size tends to infinity. It converges to a random fractal, the holomorphic multiplicative chaos. We describe this process on the unit circle, and show how it can be connected even more strongly to random matrices, and how magic square combinatorics are a type of ‘signature’ of this holomorphic multiplicative chaos. We’ll review some open questions about these objects, and discuss some links between this and other stochastic processes such as the Gaussian multiplicative chaos, the circular beta-ensemble and random multiplicative function.

The Coulomb gas is a statistical physics model consisting of N particles interacting with electrostatic repulsion and with a confining potential. I will first review results on the microscopic structure on the gas. Then, I will show how a certain subharmonic structure associated with the k-point correlation function arises. This structure implies new bounds on quantities such as the furthest particle from the origin while generalizing bounds known for the Ginibre ensemble, and it also explains how Poisson point process statistics take over in the high-temperature regime.

We give geometric conditions which imply that the space obtained by coning off the boundary components of a hyperbolic manifold $M$ is negatively curved.  Moreover, we give explicit geometric conditions under which a locally convex subset of $M$ gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic of cohomological dimension $n$ and the $\pi_1$-image of the coned-off locally convex subset is a quasi-convex subgroup. This is joint work with Jason Manning.