Mohammed Abouzaid
Title: Bordism valued GW invariants
Abstract: The geometric input of Gromov-Witten theory are moduli spaces of (pseudo)-holomorphic curves with target a (closed) symplectic manifold. It has long been known that these are not in general manifolds, because of the presence of symmetries, and that they are not in general orbifolds either, since symmetries can obstruct transversality. One model for the structure they carry is that of derived orbifolds. This motivates the study of the bordism groups of stably complex derived orbifolds as a universal receptacle for Gromov-Witten invariants in symplectic topology. I will describe joint work with Shaoyun Bai, which uses the functoriality of the resolution of singularities algorithm for complex algebraic varieties, together with refinements of Fukaya and Ono's old idea of normally complex perturbations, to split the inclusion of the bordism group of stably complex manifolds (unitary bordism) into this mysterious group, and thus proving the existence of well-defined GW invariants valued in complex cobordism groups.
Alexey Basalaev
Title: Mirror symmetry for affine cusp singularities
Abstract: In this talk we will consider the Landau-Ginzburg orbifolds given by affine cusp singularities with the non-maximal symmetry groups. We will build the Dubrovin-Frobenius manifold associated to such a Landau-Ginzburg orbifold and prove the mirror symmetry isomorphism to the Gromov-Witten theory of the Geigle-Lenzing lines.
Alexey Bondal
Title: Mirror symmetry for minuscule varieties
Abstract: We will discuss the minuscule varieties, the orbits of highest vectors in minuscule representations of complex reductive Lie groups. Then we will describe the minuscule descent, a procedure of transfer from minuscule variety to a minuscule variety for another reductive group, which has many appearance in mathematics and physics. Then we describe toric degenerations and mirror symmetry for minuscule varieties. This is a joint work with S. Galkin.
Christopher Brav
Title: Non-commutative calculus, connections, and loop space
Abstract: The Cartan calculus concerns vector fields on a smooth variety X acting on differential forms via Lie derivative and contraction, with Cartan's magic formula expressing the relation between the two actions: Lie derivative is the graded commutator of the de Rham differential with contraction. On a smooth variety, the magic formula can be checked in local coordinates, while for singular schemes (and more general prestacks) it is necessary to work with the tangent complex, where it is no longer feasible to give explicit local formulas. Interpreting the magic formula as giving Griffiths transversality for the Gauss-Manin connection of the universal infinitesimal deformation of X, we are able to construct a formula-free, chain level Cartan calculus using the tangent complex of a singular scheme, and to establish the compatibility of this calculus with the noncommutative calculus of Hochschild cochains acting on Hochschild chains.
Cheol-Hyun Cho
Title: Mirrors of invertible curve singularities via Floer theory
Abstract: Berglund-Hübsch mirror symmetry is a duality between two invertible polynomials and their symmetry groups. Given an invertible curve singularity, we explain how to construct its mirror transpose polynomial intrinsically via Floer theory. This enables us to define a canonical functor that takes curves in the Milnor fiber of one singularity to the matrix factorizations of its mirror polynomial. This is a joint work with Choa and Jeong.
William Donovan
Title: Derived symmetries for crepant resolutions of hypersurfaces
Abstract: Given a singularity with a crepant resolution, a symmetry of the derived category of coherent sheaves on the resolution may often be constructed using the formalism of spherical functors. I will introduce this, and new work (arXiv:2409.19555) on general constructions of such symmetries for hypersurface singularities. This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.
Bohan Fang
Title: Fourier transform of the central charge
Abstract: The Fourier transform of the equivariant quantum cohomology central charge, in principle, gives the quantum cohomology central charge of symplectic quotient. The process transforms the equivariant parameters to the Kahler parameters in the quotient. I will explain such transforms in the toric GIT setting and in its mirror LG model. Starting from the elementary computation on C^n, this process gives an expression of the (big) quantum cohomology central charge for toric Fano orbifolds in terms of oscillatory integrals. This talk is based on the joint work with Konstantin Aleshkin and Junxiao Wang.
Shuai Guo
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.
Wahei Hara
Title: On the non-existence of tilting objects and noncommutative resolutions
Abstract: During this talk we discuss the existence of tilting objects and noncommutative resolutions for various singularities.
The first half of the talk will discuss the topological obstruction theorem for tilting objects, which is the main theorem of this talk.
As applications of the main theorem, the second half will provide examples including
(1) threefold crepant resolutions that admit no tilting object, and
(2) higher dimensional flops in which the DK conjecture holds but tilting objects do no exist This is all joint work with Michael Wemyss.
Jianxun Hu
Title: Gamma conjecture I: counterexamples and modifications
Abstract: Gamma conjectures were proposed by Galkin, Golyshev and Iritani, and consist of conjecture O, Gamma conjecture I and II. In this talk, I will introduce the original conjectures and some counterexamples for conjecture O and Gamma conjecture I. Then I will talk about some modifications for Gamma conjecture I. Finally, I will discuss an interplay of birational transformations with an extension of Gamma conjecture I over Kahler moduli space.
Maxim Kazarian
Title: xy swap duality in topological recursion
Abstract: The topological recursion or the Chekhov-Eunard-Orantin recursion is an inductive procedure allowing one to solve in a uniform way many enumerative problems. The initial data of recursion involves two meromorphic functions on a Riemann surface denoted usually by x and y. The xy swap relations relate solutions of two topological recursions with the roles of the x and y functions swapped. The very existence of such relations implies numerous applications clarifying the nature of topological recursion, in particular:
- it leads to explicit closed formulas for the resulting differentials of the recursion in many cases that avoid the inductive procedure
- it allows one to extend the recursion to the case of degenerate pairs of x and y functions and to analyze the dependence of the resulting differentials on x and y functions
- it explains KP integrablilty property observed in many enumerative problems
The talk is based on a series joined papers with A.Alexandrov, B.Bychkov, P.Dunin-Barkowsky, S.Shadrin.
Sukjoo Lee
Title: Irregular Hodge filtration, Double covers, and Mirror symmetry.
Abstract: The Hodge filtration on the twisted cohomology of a Landau–Ginzburg model—particularly when the pole divisor is non-reduced—is known as the irregular Hodge filtration, introduced by Jeng-Daw Yu. This filtration also arises naturally in mirror symmetry and induces a generalization of the usual symmetry of Hodge numbers. In this talk, I will present two mirror symmetry examples where this filtration plays a central role: (1) stacky Clarke mirror pairs, and (2) singular double cover Calabi–Yau mirror pairs constructed by Hosono–Lee–Lian–Yau. This is a joint work with Andrew Harder.
Xiaobo Liu
Title: Universal Equations for Gromov-Witten Invariants
Abstract: Relations among tautological classes on moduli spaces of stable curves have important applications in cohomological field theory. For example, relations among psi-classes and boundary classes give universal equations for generating functions of Gromov-Witten invariants of all compact symplectic manifolds. In this talk, I will talk about such relations and their applications to Gromov-Witten theory and integrable systems.
Kaoru Ono
Title: Lagrangian Floer theory on symplectic orbifolds
Abstract: Based on joint work with Bohui Chen and Bai-Ling Wang, I will discuss Lagrangian Floer theory on symplectic orbifolds, in particular, dihedral twisted sectors as well as some issues.
Nathan Priddis
Title: Seiberg-like duality for resolutions of determinantal varieties
Abstract: 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.
Victor Przyjalkowski
Title: Exceptional collections, rationality, and fibers of Landau-Ginzburg models
Abstract: We discuss a relation of exceptional collections in derived categories of smooth Fano varieties and singularities of their Landau–Ginzburg models. We also relate rationality of Fano varieties and a monodromy of their Landau–Ginzburg models. The main example for the talk will be Picard rank one smooth Fano threefolds.
Mauricio Romo
Title: Some categorical aspects of monodromies of Calabi-Yau flops
Abstract: I will present some recently found results on the action of autoequivalences of the derived category of Calabi-Yau (CY) 3-folds. These results are based on interpreting a subset of the autoequivalences as elements of the fundamental group of the quantum Kahler moduli. In particular, I will focus on certain classes of flops and show how mondromies can be decomposed in simpler components. This is based on joint work with B. Lin.
Yongbin Ruan
Title: Counting curves in Calabi-Yau 3-folds
Kyoji Saito
Title: The semi-infinite Hodge filtration and primitive forms for hyperbolic root systems
Abstract: We construct the semi-infinite Hodge filtration and the primitive forms associated to a hyperbolic root systems of rank 2. Then, we compare the period domain with the space of stability conditions for the corresponding CY-category.
Emanuel Scheidegger
Title: Derived Categories and Enumerative Invariants of Genus One Fibrations
Abstract: We study genus one fibered Calabi-Yau threefolds from the point of view of gauged linear sigma models. In the case of 5-sections this predicts the existence of homologically projective dual pairs. In the case of 6-sections this predicts equivalences with a particular class of noncommutative threefolds determined by the presence of a B-field flux. In either case, this analysis can be extended to predictions for higher genus Gopakumar-Vafa invariants.
Junwu Tu
Title: Geometry from Categorical Enumerative Invariants
Abstract: In this talk, we discuss the problem of extracting geometric structures on moduli spaces of Calabi-Yau 3-folds from the B-model categorical enumerative invariants. Roughly speaking, we shall see that the genus zero part is essentially the underlying variation of Hodge structures; the genus one part may be packaged as a holomorphic connection on the canonical bundle of the Calabi-Yau moduli space; while the higher genus part yields a D-module structure. Such structures were previously proposed by Costello, Kontsevich-Soibelman, both inspired by Witten’s interpretation of the holomorphic anomaly equation.
Kazushi Ueda
Title: On the Fukaya categories of projective hypersurfaces of general type
Abstract: We discuss the relation between Fukaya categories of affine and projective hypersurfaces from two points of view. One is that of deformations, and the other is that of functors defined by Lagrangian correspondences. Homological mirror symmetry for projective hypersurfaces whose degree is sufficiently large follows as a corollary.
Di Yang
Title: Orbifold Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$ and topological difference equations
Abstract: Let $(m1,m2)$ be a pair of positive integers. We introduce a matrix linear difference equation, prove existence and uniqueness of its formal Puiseux-series solutions, and use them to give conjectural formulas for $k$-point ($k\ge2$) functions of orbifold Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$. Explicit expressions of the unique solutions are also obtained. These results generalize the case with $m_1=m_2=1$. We carry out concrete computations of the first few invariants by using the conjectural formulas. For the case when one of $m_1, m_2$ equals 1, we prove validity of the conjectural formulas with $k\ge3$. The talk is based on a joint work with Zhengfei Huang.
Yang Zhou
Title: Quasimap wall-crossing, generalizations and applications
Abstract: Gromov-Witten theory counts curves in a smooth projective manifold via intersection theory on the moduli of stable maps. The theory of quasimaps provide a family of alternative compactifications of the mapping space, depending on a stability parameter epsilon. The space of stability conditions is divided into chambers and the invariants are related by wall-crossing formulas. In this talk I will discuss quasimap wall-crossing via master space technique, and then I will talk about some generalizations and applications.