Schedule of talks:

See also the Arithmetic Geometry Seminar

Abstracts

(Engel): Elliptic surfaces fibering over an elliptic curve with 12 singular fibers form a ten-dimensional moduli space. Their middle cohomology has a K3 type Hodge structure, endowing the moduli space with a period map into a ten-dimensional arithmetic quotient. I will discuss a proof that this period map is dominant (like moduli of K3 surfaces), but that the degree of the period map exceeds one (unlike moduli of K3 surfaces). This is joint work with F. Greer and A. Ward.

(Gorsky): I will introduce and discuss a remarkable class of algebraic varieties, called braid varieties. These include all open Richardson and positroid varieties, and are closely related to augmentation varieties for Legendrian links. The topology of braid varieties is related to various link invariants such as HOMFLY polynomial and Khovanov-Rozansky homology, while their coordinate ring has a cluster structure. The talk is based on joint works with Roger Casals, Mikhail Gorsky, Ian Le, Linhui Shen and Jose Simental.

(Mason): The study of matroids and their invariants has undergone remarkable developments in recent years. In particular, many long-standing conjectures concerning inequalities that are satisfied between certain invariants, such as the number of flats of a given rank, associated to a given matroid, have been resolved. We take a different perspective and consider inequalities between invariants of different matroids. By employing combinatorial and algebraic methods, we prove several results, with our main result being that the flag h-vector is nonincreasing under weak maps. The talk is intended to be accessible to graduate students.

(Banerjee): In this talk I will survey what is known about the monodromy group of all smooth algebraic curves embedded in an algebraic surface X of a fixed divisor class |D|. I will discuss how π₁X can present obstructions to the monodromy group being finite index in the mapping class group. If time permits, I will discuss how the image of the monodromy group in quotients of the mapping class group by Johnson subgroups are ``as big as possible'' assuming that D is sufficiently ample.

(Jaramillo Puentes): Motivic homotopy theory allows us to tie together the results from classical and real enumerative geometry, and yield invariant counts of solutions to geometric questions over an arbitrary field k. The enumerative counts are valued in the Grothendieck-Witt ring GW(k) of nondegenerate quadratic forms over k and we call it quadratic enrichment. In this talk, I will detail some examples of these counts and I will present a quadratically enriched version of the Bernstein-Khovanskii-Kushnirenko theorem, as well as a quadratically enriched version of the Correspondence Theorem for counting curves passing through configurations of k-rational points and allowing for computations of arithmetic Gromov-Witten invariants.

(Betts): The Neron-Ogg-Shafarevich criterion asserts that an abelian variety over Q_p has good reduction if and only if the Galois action on its Z_l-linear Tate module is unramified (for l different from p). In 1995, Oda formulated and proved an analogue of the Neron-Ogg-Shafarevich criterion for smooth projective curves X of genus at least two: X has good reduction if and only if the outer Galois action on its pro-l geometric fundamental group is unramified. In this talk, I will explain a relative version of Oda's criterion, due to myself and Netan Dogra, in which we answer the question of when the Galois action on the pro-l torsor of paths between two points x and y is unramified in terms of the relative position of x and y on the reduction of X. On the way, we will touch on topics from mapping class groups and the theory of electrical circuits, and, time permitting, will outline some consequences for the Chabauty-Kim method.

(Oprea): I will discuss recent results describing the Chow groups and the tautological classes of the moduli space of quasi-polarized K3 surfaces of degree 2. This is based on joint work with Rahul Pandharipande and Samir Canning.

(Kennedy): Three seemingly different situations lead to the same construction: (1) compactifying curvilinear data (algebraic geometry), (2) studying Goursat distributions (differential geometry), and (3) analyzing a truck with trailers (dynamics and control theory. I will explain the connections among them, and show how the notion of singularity arises in each situation. To conclude, I will present a newly-discovered formula describing the slow growth sequence of a Goursat distribution. This is joint work with Corey Shanbrom and Susan Colley.

(Dennin): Gaetz and Gao recently proved the strong Sperner property for weak order by extending a rank-lowering operator ∇, first introduced by Stanley, to an sl(2) poset representation. This was done by explicitly constructing the corresponding raising operator Δ. Hamaker, Pechenik, Speyer, and Weigandt later showed that ∇ (and hence Δ) can be realized certain differential operators acting on Schubert polynomials - polynomial representatives for Schubert cycles in the cohomology of the flag variety. We outline new bijective proofs of these derivative identities for Schubert polynomials using the combinatorics of pipe dreams. These proofs extend to give related identities for beta-Grothendieck polynomials.

(Bejleri): The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing. I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

(Sutherland): The desire to solve polynomials has a lengthy, complicated history. Modern approaches to this problem center a spectrum of notions of complexity in algebra and geometry; the ends of this spectrum are known as essential dimension and resolvent degree. In this talk, we will take a tour through this mathematical landscape, including some of the relevant history which has led us to these frameworks. We will conclude by examining what we know (and what we don't) about resolvent degree, including comparisons with similar statements regarding essential dimension.

(Arguz): The KSBA moduli space, introduced by Kollár--Shepherd-Barron, and Alexeev, is a natural generalization of "the moduli space of stable curves" to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+\epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X and an anticanonical divisor D, with B=D+ε C where C is an ample divisor, it was conjectured by Hacking-Keel-Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory, mirror symmetry and punctured log Gromov-Witten theory.

(Bousseau): Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. They play an important role in the description of BPS states of supersymmetric quantum field theories. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Hülya Argüz (arXiv:2302.02068 and arXiv:2308.07270).

(Schock): The moduli space of cubic surfaces together with the labeled (marked) sum of their 27 lines is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth century work of Cayley and Salmon. We describe the natural compactifications of these spaces by KSBA weighted stable pairs, generalizing work of Hacking-Keel-Tevelev and Gallardo-Kerr-Schaffler. We additionally discuss several aspects of the geometry of these moduli spaces, including explicit descriptions of their Chow and cohomology rings, and their cones of W(E₆)-invariant effective and nef divisors. Our techniques combine classical algebraic geometry and birational geometry with tropical geometry and combinatorics of root systems.

Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2022-2023

Ohio State University Algebraic Geometry Seminar-Year 2021-2022

Ohio State University Algebraic Geometry Seminar-Year 2020-2021

Ohio State University Algebraic Geometry Seminar-Year 2019-2020

Ohio State University Algebraic Geometry Seminar-Year 2018-2019

Ohio State University Algebraic Geometry Seminar-Year 2017-2018

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

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