Joint with Differential Geometry Seminar

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general audience in mathematics and related fields.

 In this talk, we consider the largest singular value of the so-called separable covariance matrix Y=A^{1/2}XB^{1/2}, where X is a random matrix with i.i.d. entries and A, B are deterministic covariance matrices (which are non-negative definite symmetric). The separable covariance matrix is commonly used in e.g. environmental study, wireless communications and financial study to model sampling data with spatio-temporal correlations. However, the spectral properties of separable covariance matrices are much less known compared with sample covariance matrices.  

Recently, we prove that the distribution of the largest singular value of Y converges to the Tracy-Widom law under the minimal moment assumption on the entries of X. This is the first edge universality result for separable covariance matrices. As a corollary, if B=I, we obtain the edge universality for sample covariance matrices with correlated data and heavy tails. This improves the previous results for sample covariance matrices, which usually assume diagonal A or high moments of the X entries. The core parts of the proof are two comparison arguments: the Lindeberg replacement method, and a continuous self-consistent comparison argument.

Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of certain neurons in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically represented as a Venn diagram-like figure called an Euler diagram. Significant progress has been made recently by recasting this problem in terms of polynomials and using tools from commutative algebra. In particular, we will describe the toric ideal of a code and a special generating set, called the universal Gröbner basis, which contains an astounding amount of information about the ideal.

We will pay special attention to two infinite classes of combinatorial neural codes. For each code, we explicitly compute the universal Gröbner basis of its toric ideal. These computations allow one to compute the state polytopes of the corresponding toric ideals, which encode all of the distinct initial ideals arising from weight orders. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

Recently, nonstandard and ultrafilter methods have been used to obtain a number of significant results in Combinatorial Number Theory.  In this talk I will provide a brief overview of some recent work in this area, focusing on the use of nonstandard methods in problems involving the existence of various types of structured sets contained in subsets of the natural numbers that satisfy various density conditions.