Marcus Michelen (UIC)
How often is a random symmetric matrix singular?
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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(1 + o(1))}. 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 a recent work which shows the first exponential upper bound on this probability. Along the way, I'll discuss 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.
