Seminar

"Counting cubic surfaces" - Anand Patel

Abstract: The moduli space of cubic surfaces is 4-dimensional, and hence in a 4-dimensional family of such surfaces we expect to find the general cubic finitely many times. Using equivariant geometry I will describe a general formula which, when evaluated over most families, computes this number. This is joint work with Anand Deopurkar and Dennis Tseng.

"Residue field summands of syzygies via canonical resolutions" - Claudia Miller

Abstract: In joint work with Michael DeBellevue, we give extensions of results of Dao and Eisenbud showing the existence of direct summands isomorphic to the residue field in all high syzygies of any module whenever the ring satisfies a certain condition. This implies that some amount of linearity is always present in the resolutions. Our method of proof is via the relative bar resolution of Iyengar and gives some idea of why these summands should exist in such abundance and why they appear from a certain degree onward. In addition, we go on to find an exponential number of these explicitly in the Golod setting using instead the bar resolution formed from A-infinity resolutions due to Burke.