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. |
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| 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. |
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| 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! |
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| 2/17 | Hannah Larson | Tautological classes and stability for strata of differentials | Tautological classes are certain natural cohomology classes on moduli spaces, whose existence follows from the very definition of the moduli problem. Famously, the tautological classes on moduli spaces of smooth curves with marked points \(M_{g,n}\) freely generate the stable cohomology as the genus tends to infinity. In this talk, I'll explain an analogue of this result for certain natural subvarieties of \(M_{g,n}\) called strata of differentials. This is joint work with Dawei Chen. |
| 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. |
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| 3/3 | Cameron Chang | Ample Vector Bundles and Schur Positivity | In this expository talk, I’ll discuss a beauitful result due to Fulton and Lazarsfeld in 1983. Given an ample vector bundle, one can ask which polynomials in its Chern classes have positive degree. Surprisingly, the answer to this involves Schur polynomials. I’ll outline a proof and discuss Schur positivity implications in combinatorics. I’ll also highlight some interesting algebro-geometric applications. |
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| 3/10 | No talk due to colloquium | ||
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| 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 |
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| 4/7 | Speaker 1 | Title 1 | Abstract 1 |
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| 4/14 | Xiangru Zeng | Title 1 | Abstract 1 |
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| 4/21 | Javier Gonzalez Anaya | Title 1 | Abstract 1 |
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| 4/28 | Luca Battistella | Title 1 | Abstract 1 |
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