Schedule and abstracts
Workshop schedule
Lectures will be held in DRL (David Rittenhouse Laboratory).
On Friday, we will be in Auditorium A4 on the first floor.
On Saturday and Sunday, we will be in Auditorium A1 on the first floor.
Breakfast and tea breaks will be held in the corridor outside of our lecture rooms.
Friday, April 7
4:00-5:00 Xiaolei Zhao
5:00-5:30 Coffee break
5:30-6:30 Dragos Oprea
7:00 Conference dinner
Saturday, April 8
8:30-9:30 Breakfast
9:30-10:10 Daniel Halpern-Leistner 1
10:10-10:30 Coffee break
10:30-11:10 Daniel Halpern-Leistner 2
11:10-11:30 Coffee break
11:30-12:30 Yukinobu Toda
12:30-2:30 Lunch break
2:30-3:15 Short talks by early career participants
2.30-2.45: Shubham Sinha (UC San Diego)
2.45-3.00: Shengxuan Liu (Warwick)
3.00-3.15: Weite Pi (Yale)
3:15-4:00 Coffee break + Poster session
4:00-4:45 Short talks by early career participants
4.00-4.15: Deniz Genlik (Ohio State)
4.15-4.30: Pat Lank (University of South Carolina)
Sunday, April 9
8:30-9:30 Breakfast
9:30-10:30 David Treumann
10:30-11:00 Coffee break
11:00-12:00 Melissa Liu
Titles and abstracts
Daniel Halpern-Leistner, Cornell University
Title: Infinite dimensional GIT and the enumerative geometry of quotient stacks
Abstract: Over the last few years, we have made some progress studying moduli problems in algebraic geometry from an intrinsic perspective, using the theory of algebraic stacks. There are now general ways to formulate a notion of semistability. We also have necessary and sufficient criteria for the existence of moduli spaces and Theta-stratifications, the latter of which are structures analogous to the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles over a Riemann surface. In terms of methods for applying these criteria in specific examples, however, these are the early days.
After briefly summarizing the general theory, I will discuss how to apply it in two examples related to enumerative geometry: the moduli of maps from a fixed Riemann surface to a quotient stack, and the moduli of maps from nodal curves to the stack BGL_n. Both projects are joint with Andres Fernandez Herrero. Studying these moduli problems has led to an infinite dimensional analog of geometric invariant theory, which I expect will be useful for several other moduli problems of this kind.
Melissa Liu, Columbia University
Title: Open/closed correspondence and mirror symmetry
Abstract: Open/closed correspondence proposed by Mayr and Lerche-Mayr relates genus-zero open Gromov-Witten invariants counting holomorphic disks in toric Calabi-Yau 3-folds to closed Gromov-Witten invariants counting holomorphic spheres in toric Calabi-Yau 4-folds. Buryak-Clader-Tessler established an open/closed correspondence between open r-spin theory and closed extended r-spin theory. I will report recent progress on open/closed correspondence based on joint work with Song Yu and joint work in progress with Konstantin Aleshkin.
Dragos Oprea, UC San Diego
Title: The moduli space of quasi-polarized K3 surfaces of degree 2
Abstract: I will discuss recent results about 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.
Yukinobu Toda, Kavli IPMU
Title: Categorical DT/PT correspondence for local surfaces
Abstract: The DT/PT correspondence is a formula which relates Donaldson-Thomas invariants counting curves in Calabi-Yau 3-folds and Pandharipande-Thomas invariants counting stable pairs on them. In the case of local surfaces, i.e. the total spaces of canonical line bundles on surfaces, I introduced dg-categories which categorify these invariants, called DT/PT categories, based on Koszul duality and singular support quotients. In this talk, I give semiorthogonal decompositions of DT-categories for reduced curve classes which categorify the DT/PT correspondence in this case. The notion of ‘quasi-BPS categories’ and categorical Hall products play important roles. This is a joint work with Tudor Padurariu.
David Treumann, Boston College
Title: Deligne-Lusztig varieties as moduli spaces of sheaves
Abstract: Deligne-Lusztig theory organizes most of the irreducible characters of a finite group G of Lie type of into "series," that are indexed by conjugacy classes of maximal abelian subgroups T of G. The representations in one series are those that appear in the cohomology of an F_p-bar-variety X equipped with an action of the finite group G x T. A basic result of Deligne and Lusztig is "orthogonality", which tells e.g. that representations in the series corresponding to T are different from representations in the series corresponding to T-prime, when T is not conjugate to T-prime. It is proved by analyzing a stratification of the quotient (X times X-prime)/G. I will explain how the varieties X and (X times X-prime)/G, and this stratification, arise as moduli spaces of constructible sheaves on a topological annulus. They have a lot in common with moduli spaces of connections on C^* with irregular singularities at zero and infinity.
Xiaolei Zhao, UC Santa Barbara
Title: Moduli spaces of stable objects in Enriques categories
Abstract: Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids. In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category. This is a joint work with Alex Perry and Laura Pertusi.
Short talks:
Deniz Genlik, Ohio State
Title: Holomorphic Anomaly Equations For C^n/Z_n
Abstract: Physics approach to higher genus mirror symmetry predicts that Gromov-Witten potential of a Calabi-Yau threefold should satisfy certain partial differential equations; namely, the holomorphic anomaly equations. Recently, by works of Lho-Pandharipande, these equations are mathematically proved for some Calabi-Yau threefolds including C^3/Z_3. After their works, many others proved holomorphic anomaly equations for various three-dimensional targets. We generalized the work of Lho-Pandharipande on C^3/Z_3 and proved holomorphic anomaly equations for C^n/Z_n for n greater than or equal to 3, which is a result beyond the consideration of physicists. This is a joint work with Hsian-Hua Tseng.
Pat Lank, University of South Carolina
Title: High Frobenius pushforwards generate the bounded derived category
Abstract: This talk concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of D^b Coh(X) the Frobenius pushforward F ^e_*G generates the bounded derived category for whenever p^e is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.
Shengxuan Liu, Warwick
Title: Kernels of categorical resolutions of nodal singularities
Abstract: Resolution of singularities is a central topic of algebraic geometry. Hironaka showed that the resolution of singularities exists over \mathbb{C}. An analogous definition in derived categories was proposed by Lunts and the existence of categorical resolutions was shown by Kuznetsov and Lunts. One thus considers whether there is a link between categorical resolutions and classical resolutions of singularities. In this talk, I will discuss the case of nodal singularities. I will start with definitions related to categorical resolutions. Then I will mention the property of the kernel generator of one categorical resolution. Also, I will describe this kernel generator explicitly in the case of the Kuznetsov component of a nodal cubic fourfold. This is joint work with W. Cattani, F. Giovenzana, P. Magni, L. Martinelli, L. Pertusi, and J. Song.
Weite Pi, Yale
Title: Refined BPS invariants for local P^2 from the moduli of one-dimensional sheaves
Abstract: This talk concerns curve counting invariants for local P^2, the total space of the canonical bundle on the projective plane. Considerations from physics predict double-indexed integral invariants for this space, called refined BPS invariants, which are expected to refine curve counting invariants from GW/DT/PT theory. Mathematically, one proposal defines BPS invariants via the perverse filtration on the cohomology of certain moduli of one-dimensional sheaves on P^2. Following this approach, we propose a conjectural asymptotic product formula for these invariants. This can be viewed as a numerical specialization of identifying two filtrations (“Perverse = Chern”) on the cohomology ring. Based on joint work with Yakov Kononov and Junliang Shen.
Shubham Sinha, UC San Diego
Short talk title: Schur bundles over Quot schemes of P^1
Abstract: I will present formulas for the Euler characteristics of Schur bundles over Quot schemes parameterizing rank r quotients of the trivial rank N bundle over P^1. Additionally, I will discuss an interesting vanishing result and its application to the quantum K-theory of Grassmannians. This is based on joint work with Ming Zhang.
Poster presentation title: Tautological bundles over Quot scheme of curves
Abstract: We find explicit formulas for the Euler characteristics of tautological bundles over punctual Quot schemes of smooth projective curve C that parameterize zero-dimensional quotients of a vector bundle E over C. The formulas suggest analogies between the Quot schemes of curves and the Hilbert scheme of points of surfaces. Our proofs rely on Atiyah-Bott localization, universality results (of Ellingsrud, Gottsche, and Lehn), and the combinatorics of Schur functions. For higher rank quotients, we obtain expressions in genus 0. This is joint work with Dragos Oprea.