Nonnegative Quadrics on Stanley Reisner Varieties

The topic of nonnegative polynomials on varieties has attracted a lot of modern interest because of their connections to optimization and real algebraic geometry. Stanley-Reisner varieties are simple varieties that are unions of coordinate planes. We will discuss the topic of nonnegative quadratic forms over Stanley-Reisner varieties and how we can classify extreme such quadratic forms. This topic is directly related to questions of positive semidefinite matrix completion and sparse semidefinite programming. We hope to explore some interesting connections between these quadratic forms and the geometry of certain associated simplicial complexes.