August 30th, 2024 – Fuxiang Yang: Asymptotic Vanishing of Syzygies
In this talk, we will be following Park’s paper on “Asymptotic Vanishing of Syzygies of Algebraic Varieties”. In particular, we will present the key ingredient of Park’s proof, a short exact sequence involving desired vector bundles which allow us to apply induction.
September 27th, 2024 – Ethan Reed: Line Bundles on the Incidence Correspondence
I will begin by discussing connections between geometry and the study of syzygies in commutative algebra. This in turn will lead to a discussion of sheaf cohomology of line bundles on flag varieties. A full description of such in positive characteristic remains elusive. I will present recent work computing character formulas for the cohomology groups in the special case of the incidence correspondence along with connections to local cohomology and the game Nim. This talk is based on joint work with Emanuela Marangone, Annet Kyomuhangi, and Claudiu Raicu.
October 4th, 2024 – Emma Dooley: An Introduction to Brill Noether Theory
Brill Noether theory is concerned with a fundamental but very subtle question: When does a smooth projective curve of genus g (nondegenerate) over C admit a map to projective space?
It is tautological that any smooth projective curve admits an embedding into some projective space, and so this question clearly becomes more interesting once we impose conditions on the curve and the map itself. Restricting to a curve C of given genus g, we ask how the answer changes as we vary C in Mg, the moduli space of genus g curves. Hence, a more precise question which I intend to address in my talk is: For what g, r and d does a general smooth curve C of genus g admit a degree d map to Pr?
In this talk I will introduce some concepts that can help us answer questions about the existence and non-existence of such Brill Noether curves.
October 18th, 2024 – Jake Zoromski: Matroids, Monomials, Massey products, and more
The minimal resolution of an ideal can be given an algebra structure in a couple of ways. One way is to tensor the Koszul complex with its quotient ring and use the Koszul complex’s natural product structure. This algebra structure tells us about resolutions over the quotient ring. When can we find “nice” generators of this algebra? When do their products vanish? What do matroids have to do with it? I will discuss results from a recent paper of mine answering these questions and more, including a classification of Golod monomial ideals in four variables.
November 1st, 2024 – Juan Ramirez: Finite Free Resolutions
I will begin by presenting the Buchsbaum-Eisenbud Structure Theorem for perfect ideals of codimension 3. I will then talk about a connection between such ideals and Linkage (Liaison) theory and also the study of Rees algebras of these ideals.