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Pierrick Bousseau (University of Georgia)
Title: The KSBA moduli space of stable log Calabi-Yau surfaces
Abstract: The KSBA moduli space of stable pairs (X,B), introduced by Kollár--Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves for higher dimensional varieties. This moduli space is described concretely only in a handful of situations. For instance, if X is a toric variety and B=D+epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. More generally, for stable pairs of the form (X,D+epsilon C) with (X,D) a log Calabi-Yau variety and C an ample divisor, it was conjectured by Hacking--Keel--Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Arguz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.
Ionut Ciocan-Fontanine (Academia Sinica)
Title: Integrality of coefficients of mirror maps (after Jockers and Mayr)
Abstract: In a 2018 Physics paper, Jockers and Mayr argued that a so-called “3-d, N=2 abelian Gauged Linear Sigma Model” leads, via the renormalization flow, to Givental’s permutation-equivariant quantum K-theory of a complete intersection in a toric variety. Among the consequences they explored was a new derivation of the integrality of the Taylor coefficients of the mirror map (for Calabi-Yau complete intersections) which leveraged the intrinsic integrality of the 3-d GLSM partition function. In this talk I will explain how the theory of quasimaps to GIT quotients can be used to justify rigorously their argument (and extend its applicability to the non-abelian case as well). This is joint work with Jorin Schug.
Ben Davison (University of Edinburgh)
Title: Okounkov's conjecture via BPS Lie algebras
Abstract: Over ten years ago, Maulik and Okounkov introduced a new class of Yangians, generalising the Yangians associated to classical semisimple Lie groups in types ADE. These Yangians are constructed from a geometric R matrix formalism built from the cohomology of Nakajima quiver varieties. Each of the Yangians is generated by a Lie algebra in the same sense that the Yangians studied by Drinfeld are generated by classical Lie algebras. Until recently, these Lie algebras remained somewhat mysterious, and in particular we did not have a formula for their graded dimensions.
Within the noncommutative DT theory of Jacobi algebras, a different construction of "BPS" Lie algebras has been developed, where we have good tools for calculating graded dimensions, as they are given by the refined BPS invariants of noncommutative 3-Calabi-Yau varieties. By showing that the positive halves of the Maulik-Okounkov Lie algebras coincide with BPS Lie algebras associated to certain quivers with potential, we can relate their graded dimensions to Kac polynomials (concerning the combinatorics of quiver representations over finite fields). In particular, this confirms a conjecture of Okounkov regarding these graded dimensions. This is joint work with Tommaso Botta.
Hiroshi Iritani (Kyoto University)
Title: Revisiting Gamma conjecture I
Abstract: Gamma conjecture I for Fano manifolds says that the limit of a certain solution (J-function) to the quantum differential equation coincides with the Gamma class. This conjecture can be interpreted as a "quantum" analogue of the index theorem, in the sense that a quantity defined analytically by rational curve counts equals a topological characteristic class (the Gamma class). In this talk, I will present new counterexamples to Gamma conjecture I and propose a revised version of the conjecture. I will also analyze the principal asymptotic class (the limit of the J-function) over the Kaehler moduli space, and explain its close connection to the birational geometry of Fano manifolds. This is joint work with Sergey Galkin, Jianxun Hu, Huazhong Ke, Changzheng Li, and Zhitong Su.
Dominic Joyce (University of Oxford)
Title: The structure of invariants counting coherent sheaves on complex surfaces
Abstract:Let X be a complex projective surface with geometric genus pg > 0. We can form moduli spaces M(r,a,k)st ⊂ M(r,a,k)ss of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M(r,a,k)ss, giving a virtual class [M(r,a,k)ss]virt in homology.
By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on.
In my paper https://arxiv.org/abs/2111.04694, in a more general context, I extended the definition of the virtual class [M(r,a,k)ss]virt to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants [M(r,a,k)ss]virt by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.
This talk will report on a project to implement this algorithm, and actually compute the invariants [M(r,a,k)ss]virt for all ranks r > 0. I prove that the [M(r,a,k)ss]virt for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with b2+ > 1 for any rank r ≥ 2.
Shuai Guo (Peking University)
Title: Genus one Virasoro constraints for Fano complete intersections in projective spaces
Abstract: The Virasoro conjecture is a concept in enumerative geometry. It states that the generating function for the Gromov–Witten invariants of a smooth projective variety is annihilated by an action of half of the Virasoro algebra. In this talk, we will first introduce a wall-crossing formula that converts heavy markings to light markings. Then, we will prove that the Virasoro conjecture for Fano complete intersections with only ambient insertions is equivalent to the Virasoro conjecture with only one ambient insertion. In the end, we will prove the Virasoro conjecture for one ambient insertion using wall crossing formula and the twisted theory. This is a work in progress with Qingsheng Zhang and Yang Zhou.
Junho Lee (University of Central Florida)
Title: A spin analog of GW/H correspondence
Abstract: The celebrated GW/H correspondence discovered by Okounkov and Pandharipande relates GW invariants of curves and Hurwitz numbers. In this talk, I will discuss a conjectural spin analog of GW/H correspondence that relates GW invariants of surfaces of general type and spin Hurwitz numbers.
Yuan-Pin Lee (Academia Sinica)
Title: Quantum K-theory of Calabi-Yau threefolds
Abstract: The aim of this talk is to explain that on a Calabi-Yau threefold a genus zero quantum K-invariant can be written as an integral linear combination of a finite number of Gopakumar--Vafa BPS invariants with coefficients from an explicit multiple cover formula. Conversely, all Gopakumar--Vafa invariants can be determined by a finite number of quantum K-invariants in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr.
This result is consistent with the “virtual Clemens conjecture” for Calabi–Yau threefolds, a weak version of which has been proved by John Parden in cohomology (but not in K-theory). If time allows, a possible extension to genus one will also be discussed.
This is a joint work with You-Cheng Chou.
Wei-Ping Li (HKUST)
Title: Higher genus Gromov-Witten invariants for Calabi-Yau threefolds.
Abstract: I will first review an effective method to study higher genus Gromov-Witten invariants of Calabi-Yau quintics using the master space technique to handle the wall-crossing from Calabi-Yau sector to Landau-Ginzburg sector. This geometric setup can be generalised to complete intersections in toric varieties in some non-trivial way. I will concentrate on the geometric nature of theory and less on the computational side, which is nevertheless an essential part of the theory.
Woonam Lim (Yonsei University)
Title: Virasoro constraints in sheaf counting theories
Abstract: Virasoro constraints, which originate from Gromov-Witten theory, has recently been transported to various sheaf counting theories. These constraints on the sheaf side can be interpreted via Joyce’s vertex algebra. In this talk, I will explain the Virasoro constraints in sheaf counting theories and some of its recent applications to the cohomology of moduli spaces of sheaves. This is based on joint works [Bojko-L-Moreira], [L-Moreira], [Kononov-L-Moreira-Pi].
Todor Milanov (IPMU)
Title: K-theoretic Heisenberg algebras and permutation-equivariant Gromov--Witten theory
Abstract: I am planning to explain first an application of the K-theoretic Heisenberg algebras of Weiqiang Wang to the foundations of permutation-equivariant K-theoretic Gromov--Witten (KGW) theory. Then I would like to explain a formula for the genus-0 permutation equivariant KGW invariants of the point which was derived using a certain integrable hierarchy of partial differential equations.
Hyeonjun Park (KIAS)
Title: Symplectic pushforwards and DT theory
Abstract: I will introduce how to pushforward shifted symplectic fibrations along base changes. This yields an étale local structure theorem for shifted symplectic derived Artin stacks via Hamiltonian reduction. One application is deformation invariance of Donaldson-Thomas invariants for Calabi-Yau 4-folds. This in particular ensures that reduced surface counting invariants detect variational Hodge conjecture. Another application is a construction of cohomological Hall algebras for 3-Calabi-Yau categories.
Francesco Sala (University of Pisa)
Title: Cohomological Hall algebras, their representations, and Nakajima operators
Abstract: In the first part of the talk, I will provide an overview of the theory of 2d cohomological Hall algebras (COHAs), focusing on the example of COHAs arising from zero-dimensional sheaves on smooth surfaces. I will also describe certain geometric representations of these COHAs and introduce Nakajima-type operators. In the second part, I will discuss a generalization and categorification of this framework. The talk is based on joint work with Diaconescu and Porta, as well as with Diaconescu, Porta, and Yu Zhao.
Gang Tian (Peking University)
Title: Symplectic Gauged Linear Sigma Model
Abstract: In this talk, I will discuss a sympectic approach to study the gauged linear sigma model. This is a program Guangbo Xu and I satarted more than a decade ago. I will present some results we have proved. I will also discuss some recent progress and applications.
Ravi Vakil (Stanford University)
Title: Bott periodicity, algebro-geometrically
Abstract: I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.
Ryo Yamagishi (University of Bath)
Title: Duality involution on symplectic moduli spaces
Abstract: The moduli space M(r,c_1,c_2) of Gieseker-semistable sheaves on a K3 or abelian surface admits a birational symplectic involution induced by taking dual sheaves when c_1=0. In this talk, I will first discuss conditions under which this involution becomes regular, and then focus on the case M(3,0,6) for a K3 surface. I will give a local description of the involution on M(3,0,6) using quiver varieties and show that the involution quotient gives rise to a new example of an irreducible symplectic variety of dimension 20 with only isolated singularities. This is a joint work with Hsueh-Yung Lin.
Tony Yu (California Institute of Technology)
Title: Decomposition of F-bundles and new birational invariants
Abstract: F-bundle is a non-archimedean version of variation of nc-Hodge structures. I will discuss the spectral decomposition theorem for F-bundles, the resulting atomic decomposition of a smooth projective variety, and new birational invariants. Part is joint with Katzarkov, Kontsevich and Pantev, part with Hinault, Zhang and Zhang.