See also the Arithmetic Geometry Seminar

Schedule of talks:

Abstracts

(Kiers): We begin by recalling some highlights from the Geometric Satake correspondence, connecting the geometry of the affine Grassmannian for a complex connected semisimple group to the representation theory of the Langlands dual group. Next, we describe three elementary applications, each of which is a new proof of a well-known result in representation theory: the Clebsch-Gordon rule, the PRV conjecture, and Wahl's conjecture. Inspired by these examples, we ask two hopeful questions: one on the existence of dense orbits in cyclic convolution varieties, and the other on a representation-theoretic relationship between two groups whose Langlands duals are embedded one inside the other.

(Cueto): Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. By contrast, their tropical counterparts have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic curve. In this talk, I will discuss recent work joint with Hannah Markwig (arXiv:2004.10891) on the combinatorics of these bitangent classes and its connection to the number of real bitangents to real smooth quartic curves characterized by Pluecker. We will see that they are tropically convex sets and they come in 40 symmetry classes. The classical bitangents map to specific vertices of these polyhedral complexes, and each tropical bitangent class captures four of the 28 bitangents. We will discuss the situation over the reals and show that each tropical bitangent class has either zero or four lifts to classical bitangent defined over the reals, in agreement with Pluecker's classification.

(Gorsky): I will construct representations of various interesting algebras (such as rational Cherednik algebras and quantized Gieseker varieties) using the geometry of parabolic Hilbert schemes of points on plane curve singularities. A connection to Coulomb branch algebras of Braverman, Finkelberg and Nakajima will be also outlined. The talk is based on a joint work with Jose Simental and Monica Vazirani.
[Lecture notes]

(Zakharov): The Jacobian of a finite graph G is a finite group, defined as the group of degree zero divisors on the vertices of G modulo linear equivalence. Kirchhoff's celebrated matrix tree theorem states that the order of the Jacobian of G is equal to the number of spanning trees. The Jacobian a metric graph Γ, defined similarly, is a real torus of dimension equal to the first Betti number of Γ, and a weighted version of Kirchhoff's theorem expresses the volume of the Jacobian of Γ as a weighted sum over all spanning trees of Γ. A recent paper of An, Baker, Kuperberg, and Shokrieh gives a geometric interpretation of the weighted matrix-tree theorem of a metric graph Γ, based on an earlier result of Mikhalkin and Zharkov. Namely, each element of Pic^g(Γ) is represented by a unique so-called break divisor. The type of break divisor defines a canonical cellular decomposition of Pic^g(Γ), and the individual terms in the volume formula for Pic^g(Γ) are the volumes of the cells. I will state and prove analogous results for the tropical Prym variety associated to a double cover of metric graphs p:Γ' → Γ, as defined by Jensen, Len, and Ulirsch. The volume of the Prym variety is calculated as a weighted sum over certain spanning cycles on the target graph Γ. The volume formula has a geometric interpretation in terms of semi-canonical representations for Prym divisors. Joint work with Yoav Len.

(Rimányi): In Schubert calculus we study the cohomology ring of Grassmannians and other geometrically relevant spaces, together with their distinguished bases of Schubert classes. Recent developments include: (a) generalizing the cohomology theory to K theory and further to elliptic cohomology, (b) studying a deformation called h-deformation motivated by Okounkov's theory of stable envelopes, (c) extending the pool of studied spaces from homogeneous spaces to Nakajima quiver varieties and further to Cherkis bow varieties. These three generalizations together reveal a new structure: 3d mirror symmetry in Schubert calculus. The talk is based on joint results with A. Smirnov, A. Varchenko, Z. Zhou, A. Weber, S. Kumar, and a work in progress with Y. Shou.

(Svaldi): One of the main goals in Algebraic Geometry is to classify varieties. The minimal model program (MMP) is an ambitious program that aims to realize this goal, from the point of view of birational geometry, that is, we are free to modify the structure of a given variety along closed subsets to improve its geometric features. According to the MMP, there are 3 building blocks in the birational classification of algebraic varieties: Fano varieties, Calabi-Yau varieties, and varieties of general type. One important question, that is needed to further investigate the classification process, is whether or not varieties in these 3 classes have finitely many deformation types (a property called boundedness). Our understanding of the boundedness of Fano varieties and varieties of general type is quite solid but Calabi-Yau varieties are still quite elusive. In this talk, I will discuss recent results on the boundedness of elliptic Calabi-Yau varieties, which are the most relevant in physics. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. This is joint work with C. Birkar and G. Di Cerbo.

(Gross): Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In joint work with Renzo Cavalieri and Hannah Markwig we define tropical psi classes as a first step towards tropical tautological ring. I will explain how we almost recover the dilation and string equations and will also point out some of the differences between the algebraic and tropical worlds.
[Lecture notes]

(Zhang): For a general cubic threefold, a vanishing cycle on a smooth hyperplane section is an integral 2-class perpendicular to the hyperplane class and has self-intersection equal to -2. The question is what is a vanishing cycle on a singular hyperplane section? We will show that there is a certain moduli space parameterizing "vanishing cycles" on all hyperplane sections and the boundary divisor answers the question. As a vanishing cycle on a smooth cubic surface is represented by the difference of two skew lines, such moduli space arises as a quotient of the Hilbert scheme of skew lines on the cubic threefold. Based on the Abel-Jacobi map on cubic threefolds studied by Clemens and Griffiths, we'll show that the moduli space is isomorphic to the blowup of the theta divisor of the at an isolated singularity.

(Fry): The tropical moduli space M_0,n^trop is a cone complex which parameterizes leaf-labelled metric trees called tropical curves. Separately, Tevelev, and Gibney and Maclagan show that M_0,n^trop is the geometric tropicalization of the classical moduli space of pointed curves M_0,n. In this talk, I will introduce a new stability condition on the moduli space given by the combinatorics of a graph Γ. I will also show that when Γ is a complete multipartite graph, the tropical moduli space M_0,Γ^trop$ is the geometric tropicalization of its classical counterpart M_0,Γ.

(Murata): In "Brunn-Minkowski inequality for multiplicities", A. Okounkov constructed a convex body (called the Newton-Okounkov body nowadays) that encodes the asymptotic behavior of multiplicities of irreducible representations, a symplectic-geometry-like result, without the tools from symplectic geometry (in particular, it is valid over a field other than C). Later, D. Anderson showed that, assuming the semigroup constructed in the construction is finitely generated, an assumption not easy to verify, the construction amounts to a toric degeneration of a projective variety together with some power of a given ample line bundle. My Ph.D. thesis reinterprets Okounkov's original construction as an iteration of degenerations to variants of normal cones (called symbolic normal cones by me). The key is a shift from the original valuation-theoretic point of view to the (asymptotic) ideal-theoretic one. In fact, the original construction takes as an input a flag of closed subvarieties and so this reinterpreted construction is actually more sensitive to the input (consequently, for example, we can expect it to be more useful in applications).

(Bossinger): The Grassmannain, or more precisely its homogeneous coordinate ring with respect to the Plücker embedding, was found to be a cluster algebra by Scott in the early years of cluster theory. Since then, this cluster structure was studied from many different perspectives by a number of mathematicians. As the whole subject of cluster algebras broadly speaking divides into two main perspectives, algebraic and geometric, so do the results regarding Grassmannian. Geometrically, the Grassmannian contains two open subschemes that are dual cluster varieties. Interestingly, we can find tropical geometry in both directions: from the algebraic point of view, we discover relations between maximal cones in the tropicalization of the defining ideal (what Speyer and Sturmfels call the tropical Grassmannian) and seeds of the cluster algebra. From the geometric point of view, due to work of Fock--Goncharov followed by work of Gross--Hacking--Keel--Kontsevich we know that the scheme theoretic tropical points of the cluster varieties parametrize functions on the Grassmannian. In this talk I aim to explain the interaction of tropical geometry with the cluster structure for the Grassmannian from the algebraic and the geometric point of view.

(Mazzon): Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that complex Calabi-Yau manifolds should come in mirror pairs, in the sense that geometrical information of a Calabi-Yau manifold can be read through invariants of its mirror. In the first part of the talk, I will introduce some geometrical ideas inspired by mirror symmetry. In particular, I will go through the main steps which relate mirror symmetry to non-archimedean geometry and the theory of Berkovich spaces. In the second part, I will describe a combinatorial object, the so-called dual complex of a degeneration of varieties. This emerges in many contexts of algebraic geometry, including mirror symmetry where moreover it comes equipped with an integral affine structure. I will show how the techniques of Berkovich geometry give a new insight into the study of dual complexes and their integral affine structure. This is based on joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.

(Berg): If a variety has no p-adic (local) points for some prime p, then it has no rational points. Conversely, the local-global principle addresses whether a variety with local points for all p must have a rational point. Counterexamples abound, and in 1971 Manin unified existing methods to detect failures of the local-global principle into a single obstruction via the Brauer group. In 2009, Skorobogatov conjectured that the Brauer group should account for all such failures for K3 surfaces. Over the last decade much work has been devoted to the study of these obstructions on K3 (and Enriques) surfaces, and more recently with a focus on understanding which elements of the Brauer group are necessary to capture these obstructions. In this talk, I will describe the geometric origin of order 3 classes in the Brauer group of certain K3 surfaces. In joint work with Varilly-Alvarado, we utilize this geometric description to show that these classes can in fact obstruct the existence of rational points. Time permitting, I'll briefly mention ongoing work with J. Petok about certain hypersurfaces in the moduli space of Enriques surfaces and their relation to Brauer classes.

(McKean): Over the complex numbers, every smooth cubic surface has 27 lines. A slightly less well-known fact (due to Schl&aulm;fli) is that over the real numbers, every smooth cubic surface must have 3, 7, 15, or 27 lines. More generally, Segre proved that the number of lines on a smooth cubic surface over any field must be 0, 1, 2, 3, 5, 7, 9, 15, or 27. We will recall Segre's geometric proof of this theorem and discuss which of these line counts are actually realized over various fields. Time permitting, we will also describe joint work with Minahan and Zhang on explicit formulas for the lines on a cubic surface in terms of three given skew lines.

Past Seminars

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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