Joe Kileel: Covering numbers and norming sets of real algebraic varieties

Speaker
Joe Kileel, UT Austin

Abstract
In this talk I will discuss two different ways to replace a real algebraic variety by “representative” finite subsets — namely, covering sets and norming sets. For covering sets, we control the number of ell_2 balls of radius epsilon needed to cover a real variety, image of a polynomial map, or semialgebraic set in Euclidean space, in terms of the degrees of the defining polynomials and number of variables. The bound improves upon the best known general bound, and its proof is more straightforward. For norming sets, we review existing upper bounds on the size of norming sets based on the Hilbert function of a real algebraic variety. Time permitting, I will present applications in randomized polynomial optimization, including low-rank tensor decompositions. This talk is based on joint works with Yifan Zhang.