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 | 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. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 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. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/10 | No talk due to colloquium | ||
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/17 | Feiyang Lin | The singular locus of splitting loci | Splitting loci are interesting degeneracy loci that arise when one has a family of vector bundles on \(P^1\). I will define splitting loci and give some examples. Building on this, I will give a precise characterization of their singular locus. The key ingredients are certain modular resolutions of singularities for splitting loci, and an understanding of the tangent map for these resolutions in some special cases. |
| Joe Harris | Rationality conditions on varieties | There are several notions of what it means for a variety to be rational, among them rationality, unirationality and rational connectivity. These coincide in dimensions 1 and 2, and indeed the class of rational varieties is central to the classical theory of curves and surfaces. In higher dimensions, however, they diverge, and there are many open problems about their behavior, which I'll describe in this talk. | |
| 3/31 | Speaker 1 | Title 1 | |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/7 | Shang Xu | Maximal Cohen-Macaulay modules on Symplectic singularities | For a singular local ring, the family of its maximal Cohen-Macaulay modules is an important characteristic. In this talk we will introduce how to construct maximal Cohen-Macaulay sheaves on a symplectic singularity by pushing forward reflexive sheaves on its crepant resolution. We will also talk about the way of constructing maximal Cohen-Macaulay sheaves on the minimal nilpotent orbits closures of simple Lie algebra by considering the characters of representations of the stabilizer group. |
| Xiangru Zeng | Toric GIT and the Secondary Fan | Geometric invariant theory, invented by Mumford (1965), is a powerful tool in modern algebraic geometry. The GIT quotient of an affine spaces by a torus $T$ gives a toric variety, and different characters of $T$ would yield different results. This phenomenon is captured by the secondary fan of Gelfand-Kapranov-Zelevinsky (1990). I will present the basic theory of affine GIT quotient alongside the combinatorics of toric GIT. I will then introduce the construction of the secondary fan and run through basic examples. If time allows, I will also mention its connection to birational geometry. | |
| 4/14 | Soyeon Kim | Unexpected toric Richardson varieties | In this talk, we introduce a new class of toric Richardson varieties, which we call unexpected toric Richardson varieties. These Richardson varieties are toric varieties with respect to the action of a torus of higher dimension (compared to the standard torus case), whose torus action was previously completely unexplored. The standard torus action on Richardson varieties and a characterization of (expected) toric Richardsons with respect to the standard torus were well understood, by the work of Kodama--Williams, Escobar, and etc. In fact, the existence of such unexpected toric Richardsons were implied by our characterization result for all toric Richardson varieties: the open Richardson variety $R_{v,w}$ in the complete flag variety is a torus $T$ if and only if the closed Richardson variety $\overline{R}_{v,w}$ is a toric variety with respect to $T$. Equivalently, this can be said that when the Bruhat interval $[v,w]$ is a lattice. Moreover, we prove that similar to the standard torus case, the moment polytopes for all toric Richardson varieties have a nice combinatorial description in terms of Bruhat interval. This is a joint work with Eugene Gorsky and Melissa Sherman-Bennett. |
| Josie Hlavinka | Homological Mirror Symmetry for Acyclic Cluster Varieties | A conjecture of Marsh, Rietsch and Williams predicts a homological mirror symmetry duality between the Grassmannian Gr(k,n) and a Landau-Ginzburg model whose underlying space is the dense open positroid cell in Gr(n-k, n). In this talk we explain the algebraic side of a new approach to this conjecture. The main components of interest are an extension of Gross-Hacking-Keel's near construction of cluster varieties as blowups of toric varieties to a genuine one of any acylic cluster variety, and interesting combinatorics involving "nearly Boolean" permutations. | |
| 4/21 | Tyler Chamberlain | Theta Characteristics of Algebraic Curves, and More! | A theta characteristic $\theta$ of a smooth curve is a square root of its canonical bundle. Mumford, extending the work of Riemann, gave an algebraic proof that $h^0(\theta)$ remains constant modulo two as the curve (and its theta characteristic) vary in a family. In this talk, we discuss Mumford’s proof and its wider implications following a paper of Joe Harris. Time permitting, we introduce the analogous setup for third roots and explain some new obstacles that arise in this situation. |
| Javier Gonzalez Anaya | Polymatroids as toric compactifications of point configuration spaces | We will discuss two novel moduli spaces of labeled points in flags of affine spaces. The first space parametrizes distinct weighted points up to translation and scaling. This construction provides a simultaneous generalization of the Hassett moduli spaces of weighted stable rational curves, as well as the Chen-Gibney-Krashen moduli space of point configurations in affine space modulo translation and scaling. The second moduli space allows points to collide freely, without any notion of equivalence between configurations. We will see that the first admits a toric compactification coinciding with the polypermutohedral variety of Crowley-Huh-Larson-Simpson-Wang, while the second is already toric and coincides with the polystellahedral variety of Eur-Larson. This is joint work with P. Gallardo and J.L. Gonzalez. | |
| 4/28 | Luca Battistella | Chow rings of moduli spaces of curves of genus one with few markings | Modular compactifications of $M_{1,n}$ parametrising only Gorenstein curves have been constructed and classified by Smyth and Bozlee-Kuo-Neff (there are a lot of them). The combinatorially simplest one $Min_{1,n}$ parametrises curves without rational tails; for $n \leq 6$, Lekili and Polishchuk identified it with a weighted projective stack or a Grassmannian. Consider the Artin stack $G_{1,n}$ of log-canonically polarised Gorenstein curves of genus one. It admits a stratification by tail type, whose strata are products of $Min_{1,m}, m \leq n$ with moduli spaces of stable rational curves. For $n \leq 6$, we find an explicit description of its integral Chow ring by patching. The Chow ring of any modular compactification (including $\overline{M}_{1,n}$) can be obtained from it by excision. Moreover, these spaces satisfy the Chow-Künneth generation property (implying rational Chow=cohomology and polynomial point count). This is joint work with Andrea Di Lorenzo. |
| Yasna Aminaei | Determinantal Varieties | Generic determinantal varieties are spaces of matrices with a given upper bound on their ranks. In this talk, we introduce their basic properties, including their description by minors, irreducibility, dimension, singular locus, etc. We then define the degeneracy locus (determinantal varieties) of maps of vector bundles. Under suitable expected codimension assumptions, these have well-behaved geometric properties. Finally, we will discuss the Porteous formula, which computes the fundamental class of a degeneracy locus in terms of Chern classes. We will conclude with some examples and applications |