Ce Ji: Toward a generalization of the Witten conjecture from spectral curve

Abstract: Over decades of development of the Witten conjecture, Many enumerative geometries are proven to be related to integrable hierarchies. Simultaneously, such theories can also be reconstructed from topological recursion, an algorithm producing multi-differential forms from the underlying spectral curve. In this talk, we propose a generalization of the Witten conjecture from spectral curve, which produce descendent potential functions related to certain reductions of (multi-component) KP hierarchy. Proof for genus zero spectral curve with one boundary will be sketched, which can be applied to deduce the rKdV integrability of deformed negative r-spin theory, conjectured by Chidambaram--Garcia-Falide--Giacchetto. 

Matthew Satriano: Beyond twisted maps: crepant resolutions of log terminal singularities and a motivic McKay correspondence

Abstract: Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. No prior knowledge of stacks will be assumed. This is joint work with Jeremy Usatine.

Xinyu Zhang: Physical approach to K-theoretic Donaldson invariants

Abstract: We consider the path integral of topologically twisted 5d N=1 SU(2) supersymmetric Yang-Mills theory on M x S1, where M is a smooth closed four-manifold. We derive the topological correlation functions, which produce K-theoretic Donaldson invariants of M.

Xinyu Zhang: Tetrahedron instantons

Abstract: In this talk, I will give an overview of tetrahedron instantons. I will describe the construction of tetrahedron instantons in string theory and in noncommutative field theory. The tetrahedron instanton partition function lies between the higher-rank Donaldson-Thomas invariants on Calabi-Yau threefolds and fourfolds, and can be computed exactly. It admits a free field representation, suggesting the existence of a novel kind of symmetry which acts on the cohomology of the moduli spaces of tetrahedron instantons. Tetrahedron instantons can also be used to study the duality between type IIA superstring theory and M-theory.

Pengfei Huang：Regular and irregular nonabelian Hodge correspondence for meromorphic G-connections

Abstract: Algebraic (integrable) connections on an algebraic  curve/variety are known to correspond to (Stokes) local systems, but a  similar statement for G-connections is not obvious. In this talk, we  will describe such a correspondence and place it within a general  framework, namely the nonabelian Hodge correspondence. The talk will  begin with a quick review of the classical theory and then proceed to  explore this nonabelian Hodge correspondence. It will end up with the  construction of moduli spaces for various filtered local systems. Based  on some recent joint work with G. Kydonakis, H. Sun and L. Zhao, and  with H. Sun.

Daniel S Halpern-Leistner: Theta-stratifications and gauged Gromov-Witten invariants I & II

Abstract: The moduli of vector bundles on a curve has a beautiful structure called the Harder-Narasimhan stratification, which has many applications to studying the geometry of this classical moduli problem. The theory of Θ-stratifications provides an intrinsic description of this stratification that generalizes to other moduli problems. Recently, with Andres Fernandez-Herrero, we use this theory, along with a newly developed method of “infinite dimensional geometric invariant theory,” to generalize the Harder-Narasimhan stratification to the moduli of maps from a smooth curve to a quotient stack V/G, where G is a reductive group and V is an affine G-variety.I will give an overview of these results. Then I will explain how to apply them in the case where V is a linear representation to give a formula for the K-theoretic gauged Gromov-Witten invariants of V in arbitrary genus. This includes some instances of quasi-maps invariants as a special case. If time allows, I will remark on the extension of these results to complete intersections.

Yingchun Zhang: A cluster algebra structure on the quantum cohomology ring of a quiver variety

In this work, we will introduce the cluster algebra and propose a cluster algebra structure on the quantum cohomology of a quiver variety. In particular, we will give a proof for the A-type cluster algebra on quantum cohomology of flag variety.This is a joint work with Weiqiang He.

Mina Aganagic：Homological link invariants from Floer theory

Abstract. A new relation between homological mirror symmetry and representation theory solves the knot categorification problem.  The symplectic side geometry side of mirror symmetry is a theory which generalizes Heegaard-Floer theory from gl(1|1) to arbitrary simple Lie (super) algebras. The corresponding category of A-branes has many special features, which render it solvable explicitly. In this talk, I will describe how the theory is solved, and how homological link invariants arise from it.

Meng-Chwan TAN: Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification

We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in [1] about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. In essence, we will relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.

Tom A Bridgeland: Geometric structures defined by Donaldson-Thomas invariants

Abstract. In recent work the author has explored the idea that the DT invariants of a three-dimensional Calabi-Yau category can be used to define a geometric structure on the space of stability conditions. The relevant structure was christened a Joyce structure and can be viewed as a kind of non-linear Frobenius structure. There are close links with hyperkahler structures and the work of Gaiotto, Moore and Neitzke, and also with recent work in physics on non-perturbative completions of partition functions. The main examples of Joyce structures considered so far involve moduli spaces of Higgs bundles and flat connections on Riemann surfaces, and are a kind of complexification of the Hitchin system. The rough plan for the three talks is (1) Introduction, and definition of Joyce structures, (2) Moduli-theoretic construction of Joyce structures on spaces of quadratic differentials, (3) Tau function associated to a Joyce structure.

Changzheng Li：Automorphisms of the quantum cohomology of the Springer resolution and applications

Abstract.In this talk, we will introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We will discuss explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit. This is based on my joint work with Changjian Su and Rui Xiong.

Todor Milanov：Reflection vectors and quantum cohomology of blowups

Abstract: Let X be a smooth projective variety with a semi-simple quantum cohomology. The monodromy group of the quantum cohomology is by definition the monodromy group of the so-called second structure connection. There is a very interesting conjecture, more or less equivalent to Dubrovin's conjecture, that gives an explicit description of the monodromy group in terms of the exceptional objects of the derived category of X. In a joint work with my student Xiaokun Xia, we made an interesting progress in proving that the conjectural description of the monodromy group is compatible with the operation of blowing up at finitely many points. More precisely, we proved that the exceptional objects in Orlov's full exceptional collection of the blowup Bl(X) that are supported on the exceptional divisor are reflection vectors. 

Weiqiang He: Equivariant Hikita conjecture for minimal nilpotent orbit

Abtract: The theory of symplectic duality is a kind of mirror symmetry in mathematical physics. Suppose two (possibly singular) manifolds are symplectic dual to each other, then there are some highly nontrivial identities between the geometry and topology of them. One of them is the equivariant Hikita conjecture. Suppose we are given a pair of symplectic dual conical symplectic singularities, then Hikita’s conjecture is a relation of the quantized coordinate ring of one conical symplectic singularity to the equivariant cohomology ring of the symplectic resolution of the other dual conical symplectic singularity. In this talk, I will focus on this case: the minimal nilpotent orbit and the slodowy slice of the subregular orbit. This is a joint work with XIaojun Chen and Sirui Yu.

Yefeng Shen: Quantum spectrum and Gamma structures for quasi-homogeneous polynomials of general type

Abstract. In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of quasi-homogenous singularities of general type. Inspired by Galkin-Golyshev-Iritani's Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for FJRW theory of general type. Here the Gamma structures are essential to understand the connection between algebraic structures of the singularities (such as Orlov's semiorthogonal decompositions of matrix factorizations) and the analytic structures in FJRW theory. The talk is based on the work joint with Ming Zhang