Random matrix theory began with the study, by Wigner in the 1950s, of high-dimensional matrices with i.i.d. entries (up to symmetry).  The empirical law of eigenvalues demonstrates two key phenomena: bulk universality (the limit empirical law of eigenvalues doesn't depend on the laws of the entries) and concentration (the convergence is robust and fast).

Several papers over the last decade (initiated by Bryc, Dembo, and Jiang in 2006) have studied certain special random matrix ensembles with structured correlations between some entries. The limit laws are different from the Wigner i.i.d. case, but each of these models still demonstrates bulk universality and concentration.

In this lecture, I will talk about very recent results of mine and my students on these general phenomena:

Bulk universality holds true whenever there are constant-width independent bands, regardless of the correlations within each band.  (Interestingly, the same is not true for independent rows or columns, where universality fails.)  I will show several examples of such correlated band matrices generalizing earlier known special cases, demonstrating how the empirical law of eigenvalues depends on the structure of the correlations.

At the same time, I will show that concentration is a more general phenomenon, depending not on the the structure of the correlations but only on the sizes of correlated partition blocks. Under  some regularity assumptions, we find that Gaussian concentration occurs in NxN ensembles so long as the correlated blocks have size smaller than N^2/log(N).

Mathematical phase retrieval is the problem of solving systems of rank-1 quadratic equations. Over the last few years, there has been much interest in constructing algorithms with provable guarantees. Both theoretically and empirically, the most successful approaches have involved direct optimization of non-convex loss functions. In the first half of this talk, we will discuss how stochastic gradient descent for one of these loss functions provably results in (rapid) linear convergence with high probability. In the second half of the talk, we will discuss a semidefinite programming algorithm that simultaneously makes use of a sparsity prior on the solution vector, while overcoming possible model misspecification.

The problem of detecting a deformation in a symmetric Gaussian random tensor is concerned about whether there exists a statistical hypothesis test that can reliably distinguish a low-rank random spike from the noise. Recently Lesieur et al. (2017) proved that there exists a critical threshold so that when the signal-to-noise ratio exceeds this critical value, one can distinguish the spiked and unspiked tensors and weakly recover the spike via the minimal mean-square-error method. In this talk, we will show that in the case of the rank-one spike with Rademacher prior, this critical value strictly separates the distinguishability and indistinguishability of the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model, arising initially from the field of spin glasses. In particular, the signal-to-noise criticality is identified as the critical temperature, distinguishing the high and low temperature behavior, of the pure p-spin model.

The random order of service (ROS) is a natural scheduling policy for systems where no ordering of customers can or should be established. Queueing models under ROS have been used to study molecular interactions of intracellular components in biology. However, these models often assume exponential distributions for processing and patience times, which is not realistic especially when operations such as binding, folding, transcription and translation are involved. We study a multi-class queueing model operating under ROS with reneging and generally distributed processing and patience times. We use measure-valued processes to describe the dynamic evolution of the network, and establish a fluid approximation for this representation. Obtaining a fluid limit for this network requires a multi-scale analysis of its fast and slow components, and to establish an averaging principle in the context of measure-valued process. In addition, under slightly more restrictive assumptions on the patience time distribution, we introduce a reduced, function-valued fluid model that is described by a system of non-linear Partial Differential Equations (PDEs). These PDEs, however, are non-standard and the analysis of their existence, uniqueness and stability properties requires new techniques.

In this talk, we present some results on the existence of weak-solutions of the Navier-Stokes equation perturbed by transport-type rough path noise with periodic boundary conditions in dimensions two and three. The noise is smooth and divergence free in space, but rough in time. We will also discuss the problem of uniqueness in two dimensions. The proof of these results makes use of the theory of unbounded rough drivers developed by M. Gubinelli et al.

As a consequence of our results, we obtain a pathwise interpretation of the stochastic Navier-Stokes equation with Brownian and fractional Brownian transport-type noise. A Wong-Zakai theorem and support theorem follow as an immediate corollary. This is joint work with Martina Hofmanov\'a and Torstein Nilssen.