Contributed Talks Schedule
Week 2
Monday 7/21 - NATRS 140:
Chair: Franco Rota
1:30 Hannah Dell
2:05 Han-Bom Moon
2:50 Angelina Zheng
3:25 Lisa Marquand
4:10 Alessandro Danelon
4:45 Yeongrak Kim
Tuesday 7/22 - NATRS 113:
Chair: Jake Kettinger
1:30 Irene Spelta
2:05 Matt Larson
2:50 Nathan Priddis
3:25 Fenglong You
4:10 Younghan Bae
4:45 Pedro Núñez
5:20 Franco Rota
Thursday 7/24 - YATES 104:
Chair: Hannah Dell
1:30 Giulio Codogni
2:05 Pim Spelier
2:50 Adrian Langer
3:25 Andrés Ibáñez Núñez
4:10 Felix Thimm
Friday 7/25 - PATH 101:
Chair: Sandra Nair
1:30 Shengxuan Liu
2:05 Xu Gao
2:50 Artan Sheshmani
3:25 Marvin Anas Hahn
4:10 Jae Hwang Lee
Titles and Abstracts
Hannah Dell. Stability conditions on free quotients
Given the data of a variety, an algebra, or more generally a triangulated category, Bridgeland stability produces a complex manifold (the space of stability conditions). What does the geometry of this manifold tell us about the starting data? In this talk we'll investigate this question by looking for so-called "geometric" stability conditions on surfaces that arise as free quotients by finite groups. This is joint work with Edmund Heng and Anthony Licata.
Han-Bom Moon. Ulrich bundles on intersection of quadrics
An Ulrich bundle is a vector bundle with very strong cohomology vanishing conditions. Eisenbud and Schreyer conjectured that every smooth projective variety possesses an Ulrich bundle. Despite many results on low dimensional varieties and special varieties, the general existence is unknown. In this talk, I will describe recent work in progress with Kyoung-Seog Lee and Jiwan Jung on the construction of Ulrich bundles on an intersection of quadrics via derived categories.
Angelina Zheng. Tropical trigonal curves
The moduli space of algebraic curves of fixed genus g admits a well-known stratification by gonality. An algebraic curve is called d-gonal if there exists a map of degree d to the projective line, or equivalently, if there exists a linear series of degree d and dimension 1. Studying the strata of this stratification is useful in order to understand the entire moduli space, not only from an algebraic point of view but also through tropical geometry. We will define the different notions of gonality in tropical geometry and first present the case d=2, i.e. hyperelliptic tropical curves, studied by Melody Chan. In particular, in this case, as for algebraic curves, the analogous definitions of gonality in tropical geometry coincide. However, the equivalence between these definitions in tropical geometry does not hold in general. In the trigonal case (d=3), we will first see that, under the assumption of 3-edge-connectivity, the equivalence of the different definitions of gonality still holds, up to tropical modifications. This allows us to construct the moduli space of 3-edge-connected trigonal tropical curves, whose dimension coincides with that of the moduli space of trigonal algebraic curves of the same genus.
Where there are no assumptions on the edge-connectivity of the graph, the definitions are not always equivalent, but it is still possible to describe tropical curves for which they are. This is a joint work with Margarida Melo.
Lisa Marquand. Equivariant Kuznetsov components of cubic fourfolds
We consider the equivariant Kuznetsov components of a cubic fourfold with a symplectic involution. We show it is equivalent to the derived category of a K3 surface, which appears as the fixed locus of the induced involution on the Fano variety of lines. This is joint work with Laure Flapan and Sarah Frei
Alessandro Danelon. Ranks and classification of infinite-strength tensors
A tensor space is a vector space V, typically of countably infinite dimension, equipped with a multilinear form f. Two tensor spaces are isogenous if each embeds into the other. Tensor decomposition yields natural notions of rank which, in the symmetric case, exploit the strength of the form f and its derivatives.We classify isogeny classes of tensor spaces in terms of their rank and discuss a new class of representations of End(V). We will also glance at the classification of biquadratic tensor spaces with large automorphism groups. Joint work with Andrew Snowden.
Yeongrak Kim. On the rank of determinant and permanent tensors
The tensor rank of a given tensor T is the smallest number of summands when we express T as a sum of decomposable tensors. A famous work by Strassen on the matrix multiplication tensor for 2 by 2 matrices emphasizes the role of the tensor rank in complexity theory. Recently, the study of the tensor rank (and various rank notions) began to interact with various concepts in algebraic geometry including secant varieties, apolarity theory, etc. Among various tensors appearing in algebra and geometry, the determinant and the permanent are of particular interests. In complexity theory, comparing their computational complexities is one of central questions as we can find it in Valiant's permanent conjecture. In this talk, we compare their tensor ranks by considering them as tensors of order n. The key observation is an improvement of lower bounds on their tensor ranks (and border ranks) using the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. If time permits, we will also discuss the exact tensor rank (and the border rank) of the 4 by 4 determinant tensor. This work is based on a joint work with Jong In Han and Jeong-Hoon Ju.
Franco Rota. Curve contractibility via non-commutative deformations
Deciding whether a subvariety of an algebraic variety is contractible is a deep problem of algebraic geometry. Even when the subvariety is a single smooth rational curve C, the question is extremely subtle. In this talk, I will assume moreover that the ambient variety is a Calabi-Yau threefold. When C is contractible, its Donovan-Wemyss contraction algebra (which pro-represents the deformation theory of C) governs much of the geometry. Our expectation is that deformation theory not only controls contractibility but detects it, even when C is not known to contract. To investigate the deformation theory of C, we use technology developed by Brown and Wemyss to describe a local model for C.I will introduce the key ideas and tools appearing in this problem, the leading conjectures, and I will describe the (partial) results I obtained so far in collaboration with G. Brown and M. Wemyss.
Matt Larson. K-theory of wonderful compactifications
Wonderful compactifications of hyperplane arrangement complements are varieties obtained by repeatedly blowing up projective space at the strict transforms of linear subspaces. Their Chow rings were used by Huh and Katz to resolve a long-standing combinatorial conjecture, the log-concavity of the coefficients of the characteristic polynomial of a hyperplane arrangement. We study the K-rings of wonderful compactifications, and we use a Frobenius splitting argument to show that they have positivity properties. This gives new combinatorial inequalities. Joint work with Chris Eur, Shiyue Li, Sam Payne, and Nick Proudfoot.
Nathan Priddis. Seiberg-like duality for resolutions of determinantal varieties
There are two natural resolutions of the determinantal variety, known as the PAX model, and the PAXY model, resp. In this talk, I will discuss how we can realize each model as lying in a quiver bundle, I will describe how the two quivers are related by mutation, and finally give a relationship between the Gromov—Witten theory of the two resolutions via a specific cluster change of variables. This is joint work with Mark Shoemaker and Yaoxiong Wen.
Fenglong You. Relative mirror symmetry and some applications.
We consider mirror symmetry for a log Calabi--Yau pair (X,D), where X is a Fano variety and D is an anticanonical divisor of X. The mirror of the pair (X,D) is a Landau--Ginzburg model (X^\vee, W), where W is a function called the superpotential. Following the mirror construction in the Gross--Sibert program, W can be described in terms of Gromov--Witten invariants of (X,D). I will explain a relative mirror theorem for the pair (X,D) and its applications to the study of the superpotential W, the classical period of W and the mirror symmetric Gamma conjecture.
Younghan Bae. Fourier transforms and Abel-Jacobi theory
Fourier analysis is a powerful tool in analysis. In the context of abelian schemes, Fourier-Mukai transform and the weight decomposition play similar roles. For degenerate abelian fiberations, the relative group structure disappears and understanding the intersection theory leads to many interesting questions. In this presentation, I will connect Fourier transform between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, I will compute the pushforward of monomials of divisor classes on compactified Jacobians via the twisted double ramification cycle formula of all codimensions. This is a joint work with Sam Molcho and Aaron Pixton.
Pedro Núñez. Indecomposability of derived categories of hyperelliptic varieties
Hyperelliptic varieties are smooth projective varieties which are quotients of abelian varieties by finite groups acting freely and not only via translations. In this talk, we discuss the semiorthogonal indecomposability of their derived categories: we conjecture that they are always indecomposable, and verify the conjecture in many examples. The proof of the indecomposability is based on a structure theorem for their Albanese morphism. This is joint work with Pieter Belmans and Andreas Demleitner.
Irene Spelta. Families of G-curves with special properties
In this talk, we explore families of curves that admit an action by a finite group G. Using their Hodge structure, we will define and investigate "special" loci within the Torelli locus. Additionally, we will discuss the decomposability properties of a generic element of these loci.
Giulio Codogni. Vertex algebras and Teichmüller modular forms
Vertex algebras are algebraic structures coming from two dimensional conformal field theory. This talk is about their relation with moduli spaces of Riemann surfaces. I will first review some background material. In particular, I will recall that a vertex algebra is a graded vector space V with additional structures, and these structures force the Hilbert-Poincaré series of V, conveniently normalized, to be a modular form. I will then associate to any holomorphic vertex algebra a collection of Teichmüller modular forms (= sections of powers of the lambda class on the moduli space of Riemann surfaces), whose expansion near the boundary gives back some information about the correlation functions of the vertex algebra. This is a generalization of the Hilbert-Poincaré series of V, it uses moduli spaces of Riemann surfaces of arbitrarily high genus, and it is sometime called partition function of the vertex algebra. I will also explain some partial results towards the reconstruction of the vertex algebra out of these Teichmüller modular forms. Using the above mentioned construction, we can use vertex algebras to study problems about the moduli space of Riemann surfaces, such as the Schottky problem, the computation of the slope of the effective cone, and the computation of the dimension of the space of sections of powers of the lambda class. On the other hand, this construction allows us to use the geometry of the moduli space of Riemann surfaces to classify vertex algebras; in particular, I will discuss how conjectures and known results about the slope of the effective cone can be used to study the unicity of the moonshine vertex algebras. This is a work in progress with Sebastiano Carpi.
Pim Spelier. Splitting formulas for the logarithmic double ramification cycle
I'll explain a splitting formula for the (logarithmic) double ramification cycle, roughly a logarithmic Gromov-Witten invariant counting maps from curves to P^1 with specified tangency at 0 and infinity. This gives a replacement for the usual loop axiom, which is crucial to the classical theory but isn't valid logarithmically.
Adrian Langer. Projective contact log varieties
Siddarth Kannan
Felix Thimm. CY3 Wall-Crossing using Virtual Classes
Wall-crossing is used in enumerative geometry to compare various counting invariants via explicit formulas. It has emerged as a powerful tool for computations and in the study of properties of generating series of such enumerative invariants. I will present joint work with N. Kuhn and H. Liu on how to use localization of virtual classes to wall-cross invariants with descendant insertions in equivariantly Calabi-Yau 3-fold geometries.
Shengxuan Liu. Coherent Systems and Gepner-Type Stability Conditions on Quintic Surfaces
Since Bridgeland defined stability conditions on derived categories twenty years ago, these structures have found widespread applications in algebraic geometry, such as in the minimal model program for moduli spaces of stable sheaves and the construction of hyperkähler varieties. The existence of Bridgeland stability conditions is now known in many cases, including all smooth algebraic surfaces. On the other hand, it is expected that there should be a Gepner-type stability condition on hypersurfaces that is invariant under certain autoequivalence of the derived category. The existence of such stability conditions is predicted by mirror symmetry. In this talk, I will explain how coherent systems on quintic surfaces can be used to construct Gepner-type stability conditions, thereby addressing a question posed by Toda. This is a work in progress.
Xu Gao. Towards a ribbon monoidal category structure via conformal blocks of VOAs
I'll report the recent progress of an on-going joint-project (with A.Gibney, D.Krashen, and J.Liu) on an algebrogeometric approach to the ribbon monoidal category structure on conformal blocks of vertex operator algebras (VOAs). This approach aims to provide a more conceptual understanding of tensor categories of VOA-modules and aims to minimalize the dependence of transcendental ingredients. The ultimate goal is to reproduce (and generalize in a certain sense) Huang-Lepowsky-Zhang's work on vertex tensor categories to boarder bases.
Artan Sheshmani. Tyurin degenerations, Relative Lagrangian foliations and categorification of DT invariants
We discuss construction of a derived Lagrangian intersection theory of moduli spaces of perfect complexes, with support on divisors on compact Calabi Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We apply a Tyurin degeneration of the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. We show that the moduli space over the Fano 4 fold given by total space of degeneration family satisfies a relative Lagrangian foliation structure which leads to realizing the moduli space as derived critical locus of a global (-1)-shifted potential function. We construct a flat Gauss-Manin connection to relate the periodic cyclic homology induced by matrix factorization category of such function to the derived Lagrangian intersection of the corresponding “Fano moduli spaces”. The latter provides one with categorification of DT invariants over the special fiber (of degenerating family). The alternating sum of dimensions of the categorical DT invariants of the special fiber induces numerical DT invariants. If there is time, we show how in terms of “non-derived” virtual intersection theory, these numerical DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Ludmil Katzarkov, Maxim Kontsevich, recent work with Jacob Kryczka, and former work with Vladimir Baranovsky.
Marvin Anas Hahn Tropicalising hypergeometric tau-functions
Hurwitz numbers and many related enumerative invariants are known to yield solutions of integrable systems when passing to their generating series. All known examples fall into the class of so-called hypergeometric tau-functions. In this talk we show that the structural behaviour of Hurwitz numbers extends to coefficients in the expansion of all hypergeometric tau-functions. This talk is based on joint work in progress with Brian O’Callaghan and Jonas Wahl.
Jae Hwang Lee Quantum modules of Semipositive Toric Varieties
Quantum cohomology is an interesting object to both mathematicians and physicists due to its relation to string theory. Its product structure, called the quantum product, deforms the product structure of the ordinary cohomology ring via 3-pointed Gromov-Witten invariants. In a similar way, one can use 2|1-quasimap invariants, which are similar to Gromov-Witten invariants, to define such a quantum deformation. It defines no longer a product structure, but a quantum module structure, an analogue of the WDVV equations. The semipositive conjecture asks if the quantum module is the same as the Batyrev module for all smooth projectve toric semipositive varieties. In this talk, a proof will be given.
Andrés Ibáñez Núñez. Intrinsic Donaldson-Thomas theory
I will explain how to define Donaldson-Thomas invariants for (-1)-symplectic 1-Artin stacks and Euler characteristic of 1-Artin stacks using the newly developed component lattice. This generalizes Joyce-Song invariants to non-linear moduli spaces, like G-bundles. The invariants satisfy wall-crossing formuli. This is joint work with Chenjing Bu and Tasuki Kinjo.
Locations
Contributed talk sessions will happen in the following classrooms. Here is a link to the CSU campus map, and you may click on each classroom to see its location.