Schedule of talks:

Abstracts

(Xia): The notion of Bridgeland stability provides a new viewpoint on studying moduli spaces of sheaves and complexes, and there has been a lot of work in this direction on curves and surfaces. In this talk, we will study the geometric structure of the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability and wall-crossing techniques. While this is classically known from the result of Piene and Schlessinger, our method does not use explicit equations of some special ideals and should work on the study of more general moduli spaces.

(Tseng): The moment map associated to the torus action on a symplectic toric manifold X gives a Lagrangian torus fibration. Fukaya-Oh-Ohta-Ono defined invariants that count, in an appropriate sense, holomorphic disks in X whose boundaries lie on a fixed generic moment map fiber L. They used these invariants to study mirror symmetry for toric manifold X. In this talk we'll explain some effective results on these invariants for semi-Fano toric manifolds, and some extensions to toric orbifolds. This is joint work with K. Chan, C.-H. Cho, S.-C. Lau, and N. C. Leung.

(Karpman): The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich geometrical and combinatorial structure. There are a number of interesting combinatorial posets which describe the closure relations among positroid varieties, just as Young diagrams give the closure relations among Schubert varieties. In addition, Postnikov's boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig's theory of total nonnegativity for partial flag varieties, and has remarkable applications to particle physics. In this talk, we extend the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form.

(Katz): I will discuss the recent proof with Joseph Rabinoff and David Zureick-Brown that there is a uniform bound for the number of rational points on genus g curves of Mordell-Weill rank at most g-3, extending a result of Stoll on hyperelliptic curves. Our work also gives unconditional bounds on the number of rational torsion points and bounds on the number of geometric torsion points on curves with very degenerate reduction type. I will outline the Chabauty-Coleman method for bounding the number of rational points on a curve of low Mordell-Weil rank and discuss the challenges to making the bound uniform. These challenges involve p-adic integration and Newton polygon estimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

(Girivaru): The Lefschetz conjectures are attempts at generalizing the classical Grothendieck-Lefschetz and Noether-Lefschetz theorems for Picard groups of a smooth projective variety and its hyperplane sections and fit in the framework of the Bloch-Beilinson conjectures. I will talk about some results I have obtained in joint work with Deepam Patel of Purdue University.

(Jiang): Then Behrend function, introduced by K. Behrend, is a fundamental tool in the Donaldson-Thomas theory. For the moduli space of coherent sheaves on a smooth Calami-Yau threefold, the Joyce-Song formula of Behrend function identities for the coherent sheaves are the essential part for the wall crossing of the counting invariants. In this talk I will talk about the motivic version of these formulas and their role in the wall crossing of motivic Donaldson-Thomas invariants.

(Janda): Topological recursion relations are very special relations between cohomology classes of the moduli space of curves. They are also of interest in Gromov-Witten theory. Recently, Pixton's double ramification relations have given many more examples of topological recursion relations. In my talk I want to discuss Pixton's double ramification relations, and how they can be used to prove a conjecture of T. Kimura and X. Liu on topological recursion relations. This is joint (and partially ongoing) work with various subsets of E. Clader, S. Grushevsky, X. Liu, X. Wang, and D. Zakharov.

(Lin): Tropicalization is an effective way to reduce the counting of Riemann surfaces into combinatorics. In this talk, we will define an open Gromov-Witten invariants on K3 surfaces, which count the holomorphic dsics with boundaries on the special Lagrangian torus. The projection of these discs to the base of the fibration are the expected "tropical discs" under certain adiabatic limit. I will how to count these tropical discs. In particular, we will establish a correspondence between the counting of tropical discs and open Gromov-Witten invariants on K3 surfaces.

(Wang): It is a theorem of de Bruijn and Erdös that n points in the plane determines at least n lines, unless all the points lie on a line. This is one of the earliest results in enumerative combinatorial geometry. We will present a higher dimensional generalization to this theorem. Let E be a generating subset of a d-dimensional vector space. Let W_k be the number of k-dimensional subspaces that is generated by a subset of E. We show that W_k≤ W_d-k, when k ≤ d/2. This confirms a "top-heavy" conjecture of Dowling and Wilson in 1974 for all matroids realizable over some field. The main ingredients of the proof are the hard Lefschetz theorem and the decomposition theorem. I will also talk about a proof of Welsh and Mason's log-concave conjecture on the number of k-element independent sets. These are joint works with June Huh.

(Hobson): In this talk we will define and explore an infinite family of vector bundles, known as vector bundles of conformal blocks, on the moduli space M_0,n of marked curves. These bundles arise from data associated to a simple Lie algebra. We will show a correspondence (in certain cases) of the rank of these bundles with coefficients in the cohomology of the Grassmannian. This correspondence allows us to use a formula for computing "quantum Kostka" numbers and explicitly characterize families of bundles of rank one by enumerating Young tableau. We will show these results and illuminate the methods involved.

(Schnell): In the past few years, people working on the analytic side of algebraic geometry have obtained two important new results: a version of the Ohsawa-Takegoshi extension theorem with sharp estimates (Blocki, Guan-Zhou), and the existence of canonical singular hermitian metrics on pushforwards of relative pluricanonical bundles (Berndtsson, Paun, Takayama, and others). In this talk, I will explore some consequences of this work for the study of morphisms to complex abelian varieties, including the recent proof of Iitaka's conjecture over abelian varieties (Cao-Paun). The talk will be understandable without any background in analysis.

(Oprea): The Verlinde bundles are constructed over the moduli space of curves by considering relative moduli spaces of vector bundles. Their Chern characters yield a cohomological field theory. I will explain how Teleman's work on semisimple cohomological field theories can be used to derive explicit formulas for the Chern characters in terms of tautological classes.

(Brosnan): Suppose S is a smooth, complex variety containing a dense Zariski open subset U, and suppose W is a smooth projective family of varieties over U. It seems natural to ask when W admits a regular flat compactification over S. In other words, when does there exist a smooth variety X flat and proper over S containing W as a Zariski open subset? Using resolution of singularities, it is not hard to see that it is always possible to find a regular flat compactification when S is a curve. My main goal is to point out that, when dim S >1, there are obstructions coming from local intersection cohomology. My motivation is the recent preprint of Laza, Sacca and Voisin (LSV) who construct a regular flat compactification in the case that W is a certain family of abelian 5-folds over an open subset of 5 dimensional projective space. On the one hand, I'll explain how to compute the intersection cohomology in certain related examples and show that these are obstructed. On the other hand, I'll use the vanishing of the intersection cohomology obstructions implied by the LSV theorem to deduce a theorem on the palindromicity of the cohomology of certain singular cubic 3-folds.

(Chung): For a flag variety X, its K-theoretic Gromov-Witten invariants encode the arithmetic genera of families of rational curves meeting Schubert varieties in X. These invariants can be assembled into the quantum K-theory ring of X, analogous to quantum cohomology. When X is cominuscule (eg. Grassmannians) the quantum K-theory ring’s structure constants satisfy a certain relation. We will discuss this relation and its connection to sheaf Euler characteristics of (non-quantum) K-theory of X. This is joint work with A. S. Buch.

(Oberdieck): Physics predicts that curve counting invariants of elliptically fibered Calabi-Yau threefolds are governed by Jacobi forms (a generalization of modular forms). I will explain joint work with Junliang Shen to prove part of the conjectured modularity by using auto-equivalences in the derived category and wall-crossing. In good cases the remaining part of the conjecture can be obtained via Gromov-Witten theory. We discuss this strategy in two explicit examples, the elliptic fibration over P² and the product of a K3 surface and an elliptic curve.

(Urbinati): Let X be a Mori Dream Space. We construct via tropicalization a model dominating all the small Q-factorial modifications. This construction is the main tool to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on X, and hence the movable cone of X. This is a joint work with Elisa Postinghel.

(Silversmith): Through 3 general points and 6 general lines in P³, there are exactly 190 twisted cubics; 190 is a Gromov-Witten invariant of P³. Mirror symmetry is a conjecture about the structure of all Gromov-Witten invariants of a smooth complex variety (or orbifold) X. The conjecture is known for toric orbifolds and some of their complete intersections. We prove it in the case of the nontoric orbifold Sym^d(P^r). This orbifold is of particular interest because when r=2, its Gromov-Witten invariants are conjecturally related to those of the Hilbert scheme Hilb^d(P²).

(Nash): Recently, a theory for the tropicalization of subvarieties of a spherical homogeneous space G/H was developed by Tassos Vogiannou. We extend his ideas to define the tropicalization of a spherical G/H-embedding. This is a generalization of the more well-known theory for tropicalizing toric varieties by extending the tropicalization of the dense torus.

(Tseng): Around a decade ago the following four (C^*)²-equivariant theories are proven to be equivalent:
(1) Gromov-Witten theory of P¹ × C² relative to three fibers;
(2) Donaldson-Thomas theory of P¹ × C² relative to three fibers;
(3) Quantum cohomology of Hilbert schemes of points on C²;
(4) Quantum cohomology of symmetric product stacks of C².
In this talk we'll discuss these four equivalence. We'll also sketch some new development, namely higher genus extensions of these equivalences (joint work with R. Pandharipande).

(Anderson): Any covariant homology theory on algebraic varieties has an associated "operational" bivariant theory, thanks to a construction of Fulton and MacPherson. Starting with Chow homology, this is how the Chow cohomology of singular varieties is defined; Sam Payne and I developed the foundations of an analogous story for K-theory. I will describe further joint work with Richard Gonzales and Payne, in which we study the operational equivariant K-theory and Riemann-Roch transformations. Despite its abstract definition, operational K-theory has many properties which make it easier to understand than the K-theory of vector bundles or perfect complexes. This is illustrated most vividly by singular toric varieties, where relatively little is known about K-theory of vector bundles, while the operational equivariant K-theory has a simple description in terms of the fan.

(Ulirsch): The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not functions as a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.
In this talk I am going to give an introduction to these fascinating moduli spaces and report on ongoing work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this 2-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose two different ways of describing the process of tropicalization: one via logarithmic geometry in the sense of Kato-Illusie and the other via non-Archimedean analytic geometry in the sense of Berkovich.

(Rodriguez): If a variety walks though the door, then the first question you might ask is, "What is your degree?". If the variety is projective (defined by homogeneous polynomials), then these questions are answered by understanding the intersection of the variety with general linear spaces. However, often we are given a system of polynomials whose variables can be grouped together in a natural way. These multihomogeneous polynomials define multiprojective varieties. When considering these varieties, one should ask "What is your multidegree?" to capture more detailed information. In this talk, a numerical procedure called multiregeneration is presented to determine the multidegree of a variety. Illustrating examples from statistics and optimization will be shown.

(Engel): A triangulation of S² has non-negative curvature if every vertex has six or fewer triangles adjacent to it. Thurston showed that non-negative curvature triangulations correspond to lattice points in a moduli space of flat cone metrics on S². In joint work with Peter Smillie, we use an arithmetic technique of Siegel to count such lattice points. The appropriately weighted number of triangulations with 2n triangles is an explicit constant times the ninth divisor power sum of n. If time permits, I will discuss work in progress on the enumeration of triangulations with any set of specified non-zero curvatures.

(Gorsky): I will discuss recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of Hilbert schemes of points on the plane. All notions will be introduced in the talk, no preliminary knowledge is assumed. This is a joint work with Andrei Negut and Jacob Rasmussen.

Past Seminars

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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