Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
3:45: Aldo Conca: Syzygies of Koszul algebras
Koszul algebras are certain algebras defined by quadratic relations; most quadratic algebras that arise naturally are in fact Koszul. There is no finite criterion for Koszulness, however, so it is interesting to study necessary or sufficiently conditions. I will explain what Koszul algebras are, and describe special features of their syzygies.
5:00: Elizabeth Gross: Toric Ideals of Hypergraphs
The edge subring of a graph G is the monomial subalgbera parameterized by the edges of G. It’s defining ideal, commonly referred to as the toric ideal of G, has been well-studied and several historical results tell us the same beautiful story: we can understand these ideals by understanding the combinatorics of the underlying graph. A natural extension is to consider the toric ideal of a hypergraph. In this talk, we will survey well-known results on the toric ideals of graphs and explain how these concepts generalize to hypergraphs. We will end with recent results on the toric ideals of hypergraphs.