Tuesdays 4-6, Evans 939
All semesters: Spring 2025, Fall 2024 , Spring 2024, Fall 2023, Spring 2023, Fall 2022
| date | speaker | title | abstract |
|---|---|---|---|
| 1/20 | Shubham Sinha | Intersection theory of Hyperquot schemes on curves | Hyperquot schemes parameterize chains of successive quotients of a fixed vector bundle V on a smooth projective curve C. When V is trivial, Hyperquot schemes provide a compactification of the space of morphisms from C to a partial flag variety. I will present a formula for computing intersection numbers on Hyperquot schemes. Our result generalizes the Vafa–Intriligator formula, proved by Marian–Oprea, for computing intersection numbers on Quot schemes. I will also discuss conditions under which our formulas yield enumerative counts of maps to flag varieties. This talk is based on joint work with R. Ontani and W. Xu. |
| David Eisenbud | Weierstrass points and Syzygies | In 1892 Hurwitz asked whether every numerical semigroup (=subset of the non-negative integers closed under addition) appears as the Weierstrass semigroup of a point on a Riemann surface (smooth projective curve). I'll explain what Weierstrass semigroups are, and some of the tangled history of this problem, as well as a new method, based on syzygies of monomial curves, to produce non-Weierstrass semigroups. This is joint work with Frank-Olaf Schreyer. | |
| 1/27 | Cameron Chang | An Introduction to Newton-Okounkov Bodies | The goal of this talk is to describe the interplay between algebraic and convex geometry. One of the first instances of this was the Bernstein-Khovanskii-Kushnirenko theorem, which gives a relationship between the number of solutions to a system of Laurent polynomial equations and the volume of a convex polytope. In their 2009 paper, Kaveh and Khovanskii realized that this could be generalized vastly, interpreting solution counts on ANY variety as some volume of a convex body. These convex bodies, called Newton-Okounkov bodies, can be leveraged to use convex geometry to prove many results purely in algebraic geometry. We will try to outline the construction in this talk, with a focus on applications. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 2/3 | Shang Xu | Kleinian Surfaces as Nakajima Quiver Varieties | The goal of this talk is to explain Mckay correspondence using Nakajima quiver varieties. Given a finite group G in SL(2,C), one can blow up several times to obtain a resolution of the isolated surface singularity in C^/G. The intersection form on the exceptional fibers gives a Dykin diagram of ADE type. While the natural 2-dimensional representation of G has a Mckay graph which is also an ADE Dynkin diagram, it is inspiring that two diagrams coincide. Mckay came up with this observation in 1980 and Artin, Verdier published a series of papers trying to explain this. In 2001, Willian Crawley-Boevey's paper related this correspondence to the resolution defined by Nakajima quiver varieties and understood the exceptional fiber in moduli aspects. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 2/10 | Jo Hlavinka | Homological Mirror Symmetry for Acyclic Cluster Varieties | The theory of cluster varieties is a subtle analogue of toric geometry which applies to many non-toric but combinatorially interesting varieties. Homological mirror symmetry, in the form of an identification between categories of symplectic and algebraio-geometric interest, is a theorem for toric varieties. In this talk, a survey with a splash of the speaker's research, we explain the prerequisites to understanding the first known HMS results for non-trivial cluster varieties. No background in symplectic geometry is assumed! |
| Speaker 2 | Title 2 | Abstract 2 | |
| 2/17 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 2/24 | Daksh Aggarwal | Brill-Noether theory for totally ramified covers of the projective line | Given a curve C that is a degree-k cover of the projective line totally ramified at two points p and q, we can seek to understand the space of degree d line bundles on the curve with prescribed ramification at p and q. The corresponding subschemes of Pic^d(C) are called transmission loci and are parameterized via elements of the k-affine symmetric group. Transmission loci provide a refinement of the splitting loci that have recently been extensively studied for general k-gonal curves. Pflueger has conjectured analogues of the classic Brill-Noether theorem should hold for transmission loci. In this talk, I will introduce the Brill-Noether theory for general curves and general k-gonal curves, and then describe the analogues for general covers of P^1 totally ramified at two points along with some ideas about the proof technique. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/3 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/10 | Bernd Sturmfels | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/17 | Feiyang Lin | Title 1 | Abstract 1 |
| Joe Harris | Title 1 | Abstract 2 | |
| 3/31 | Speaker 1 | Title 1 | No seminar due to postdoc talks |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/7 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/14 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/21 | Javier Gonzalez Anaya | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| Date | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| Date | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| Date | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 |