Georgia Institute of Technology College of Sciences

Let X_{N,n} be an iid product of N real Gaussian matrices of size n x n. In this talk, I will explain some recent joint work with G. Paouris 
(arXiv:2005.08899) about a non-asymptotic analysis of the singular values of X_{N,n} . I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of X_{N,n} to the so-called Triangle Law in the limit where N,n tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large N on the distance between the joint distribution of all Lyapunov exponents of X_{N,n} and appropriately normalized independent Gaussians in the near-ergodic regime (N >> n).

If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdős-Rényi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.

We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.

Prompted by the definition for the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of graded rings for which this generating function is rational. We will discuss this class of rings.  This work is joint with Yongwei Yao.