Schedule of talks:

Abstracts

(Tseng): Let S be a nonsingular surface. A version of the crepant resolution conjecture predicts that the descendant Gromov-Witten theory of Hilbⁿ(S), the Hilbert scheme of n points on S, is equivalent to the descendant Gromov-Witten theory of Symⁿ(S), the n-fold symmetric product of S. In this talk we discuss how this works when S is C². We explicitly identify a symplectic transformation equating the two descendant Gromov-Witten theories. We also establish a relationship between this symplectic transformation and the Fourier-Mukai transformation which identifies the (torus-equivariant) K-groups of Hilbⁿ(C²) and Symⁿ(C²). This is based on joint work with R. Pandharipande.

(Webb): When a complex reductive group G with maximal torus T acts on an affine variety S, one can form two (GIT) quotients: W//G and W//T. With the right hypotheses, W//G and W//T are both smooth projective varieties. The relationship between the cohomology rings of these two varieties is well understood. I will discuss the relationship of their quantum cohomology using the quasimap theory of Ciocan-Fontanine and Kim. In particular, I will present the abelian-nonabelian correspondence for quasimap I-functions when W is a vector space and G is connected.

(Cueto): Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in P³ contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP³. In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P⁴⁴ via its anticanonical bundle. The combinatorics of the root system of type E₆ and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.

(Richman): The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. finite metric graphs), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.

(Hast): Given a curve of genus at least 2 over a number field, what can we say about its set of rational points? Faltings' theorem tells us that this set is finite, but many questions remain about how to obtain good bounds on the number of rational points and how to provably list all rational points. We will survey some recent progress and ongoing work on these questions using Kim's non-abelian Chabauty method, which uses the fundamental group to construct p-adic analytic functions that vanish on the set of rational points.
In particular, we present a new proof of Faltings' theorem for superelliptic curves over Q, due to joint work with Jordan Ellenberg. We will also discuss a conditional generalization of this strategy from Q-points to points in any real number field.

(Yang): Given a Fano variety X over a number field, and a height function on X. Manin's conjecture predicts the asymptotic growth of the number of rational points on X of bounded height. When X is an equivariant compactification of a homogeneous variety, the conjecture is related to equidistribution of adelic periods. We will discuss some recent progress in this direction.

(Ullery): The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. We will discuss these measures of irrationality and various methods of calculating and bounding them. We will mainly focus on the examples of hypersurfaces and, more generally, complete intersections in projective space.

(Doran): The quest to understand the commonalities and differences between topology and algebraic geometry is central to many mathematical developments. Here we consider a number of seemingly unrelated but well-known unsolved problems (in geometry, number theory, representation theory): for instance, dating back to Newton's time, what is the minimal degree curve passing through n chosen points in the plane with multiplicities? Relatedly, a common, but famously difficult, broad question is to determine when a ``homology" class is represented by a (symplectic, algebraic, etc.) geometric object, that is to say, when is a class effective?

We recast these problems using a canonical construction that crucially exploits an oft-overlooked difference between topological and algebraic structures. The resulting "uncloaked" problems can then all be attacked with the same weapon. At least up to scaling, this is a form of stability analysis using both additive and multiplicative groups. Beautiful combinatorial structures, like scuffed polytopes and modifications of Okounkov bodies arise naturally. The results are suggestive that effectivity may reduce to stability much more broadly still, and that many wall-crossings in geometry may admit interpretation as a change in this form of stability.

If there is time, we may briefly mention links with quantum entanglement and entropies, and question whether physics is sensitive to some of these differences between topology and algebraic geometry.

Some of this work is joint with Frances Kirwan.

(Denham): The maximal likelihood variety of a complex hyperplane arrangement describes the set of critical points of all rational functions with poles and zeros on the arrangement. This variety's bidegree encodes the h-vector of the underlying matroid's broken circuit complex. I will describe work-in-progress with Federico Ardila and June Huh that constructs a tropical version of the maximal likelihood variety. This sheds some new light on the tropical characteristic classes of López de Medrano, Rincón, and Shaw, as well as on the h-vector of the broken circuit complex of an arbitrary matroid.

(Anderson): A basic problem from the 19th century asks for the degree of the locus of symmetric matrices of bounded rank; answers were given by Schubert and Giambelli. More recently, many extensions of this problem have been considered, including versions coming from symmetric maps of vector bundles, or for vector bundles equipped with a nondegenerate bilinear form. I will discuss ongoing work with William Fulton, in which we allow the bilinear form to be "twisted", so that it takes values in a nontrivial line bundle. The formulas we obtain extend those of Billey-Haiman, Ikeda-Mihalcea-Naruse, and others, and exhibit new connections with algebraic combinatorics.

(Tseng): For a smooth projective variety X containing a smooth irreducible divisor D, the question of counting curves in X with prescribed contact conditions along D is a classical one in enumerative geometry. In more modern approaches to this question, there are two ways to define these counts: as Gromov-Witten invariants of X relative to D, or as Gromov-Witten invariants of the stack X_D,r of r-th roots of X along D (for r large). In genus 0, explicit calculations in examples suggested that these two sets of Gromov-Witten invariants are always the same, whether they actually count curves or not. This is proven in full generality by Abramovich-Cadman-Wise. The situation in genus > 0 is not so simple, as an example of D. Maulik showed that the two sets of Gromov-Witten invariants are not equal, even in genus 1. In this talk, we'll explain how these two sets of Gromov-Witten invariants are related in all genera, in full generality. This is based on a joint work with Fenglong You.

(Jensen): he geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

(Riedl): We discuss several related notions of hyperbolicity of projective varieties, focusing particularly on algebraic hyperbolicity and Brody hyperbolicity. We discuss what is known about these notions for very general hypersurfaces in projective space. In joint work with Coskun, we prove that quintic hypersurfaces in P³ are algebraically hyperbolic, finally settling the last case of a conjecture of Demailly. In joint work with David Yang, we show that (a slightly stronger version of) the Green-Griffiths-Lang Conjecture implies the Kobayashi Conjecture. This substantially simplifies the proof of the Kobayashi Conjecture and has led to improved degree bounds.

(Su): Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump-Nakasuji-Naruse about certain transition matrix between these two bases. The idea of the proof is to transform the problem to the Langlands dual side, and relate the Iwahori invariants to Maulik-Okounkov's stable envelopes and Brasselet-Schurmann-Yokura's motivic Chern classes for the Langlands dual flag variety. Joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

(Tseng): In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.

Past Seminars

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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