## Geometry Seminar

**Date: ** May 23, 2022

**Time: ** 3:00PM - 4:00PM

**Location: ** BLOC 302

**Speaker: **Justin Chen, ICERM Bridge

**Title: ***Sums of squares, Hankel index, and almost real rank*

**Abstract: **The difference between nonnegative polynomials and sums of squares is an important topic in real algebraic geometry, and the Hankel index of a variety is a natural (but subtle) invariant that quantifies this difference. Surprising connections were found between the Hankel index and the commutative algebra of the variety, namely the N_{2,p} property of linear syzygies for the free resolution – although this only provides half the story. For curves of almost minimal degree, we complete the picture, by determining the Hankel index in terms of a new rank called almost real rank, which interpolates between real (Waring) rank and complex border rank. This is joint work with Greg Blekherman and Jaewoo Jung.