Speaker:  Marcus Michelen (University of Illinois Chicago)

Title:  Singularity and the least singular value of random symmetric matrices

Abstract: Consider the random symmetric n x n matrix whose entries on and above the diagonal are independent and uniformly chosen from {-1,1}.  How often is this matrix singular?  A moment's thought reveals that if two rows or columns are equal then the matrix is singular, so the singularity probability is bounded below by 2^-n(l+o(l)).  Proving any sort of upper bound on the singularity probability turns out to be difficult, with results coming slowly over the past two decades.  I'll discuss work which shows the first exponential upper bound on this probability as well as extensions to more quantitative notions of invertibility.  Along the way, I'll describe some tools---both old and new---which are powerful and (hopefully) interesting in their own right.  This talk is based on joint work with Marcelo Campos, Matthew Jenssen, and Julian Sahasrabudhe.