Online Seminar

11242023
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 multidifferential 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 (multicomponent) 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 rspin theory, conjectured by ChidambaramGarciaFalideGiacchetto.

11162023
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 logterminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stacktheoretic 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.

10312023
Xinyu Zhang: Physical approach to Ktheoretic Donaldson invariants
Abstract: We consider the path integral of topologically twisted 5d N=1 SU(2) supersymmetric YangMills theory on M x S1, where M is a smooth closed fourmanifold. We derive the topological correlation functions, which produce Ktheoretic Donaldson invariants of M.

10312023
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 higherrank DonaldsonThomas invariants on CalabiYau 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 Mtheory.

10192023
Pengfei Huang：Regular and irregular nonabelian Hodge correspondence for meromorphic Gconnections
Abstract: Algebraic (integrable) connections on an algebraic curve/variety are known to correspond to (Stokes) local systems, but a similar statement for Gconnections 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.

09222023
Daniel S HalpernLeistner: Thetastratifications and gauged GromovWitten invariants I & II
Abstract: The moduli of vector bundles on a curve has a beautiful structure called the HarderNarasimhan stratification, which has many applications to studying the geometry of this classical moduli problem. The theory of Θstratifications provides an intrinsic description of this stratification that generalizes to other moduli problems. Recently, with Andres FernandezHerrero, we use this theory, along with a newly developed method of “infinite dimensional geometric invariant theory,” to generalize the HarderNarasimhan 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 Gvariety.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 Ktheoretic gauged GromovWitten invariants of V in arbitrary genus. This includes some instances of quasimaps invariants as a special case. If time allows, I will remark on the extension of these results to complete intersections.

09182023
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 Atype cluster algebra on quantum cohomology of flag variety.This is a joint work with Weiqiang He.

09082023
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 HeegaardFloer theory from gl(11) to arbitrary simple Lie (super) algebras. The corresponding category of Abranes 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.

08092023
We revisit VafaWitten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full VafaWitten equations. We physically derive (i) a novel VafaWitten fourmanifold invariant associated with this moduli space, (ii) their relation to GromovWitten invariants, (iii) a novel VafaWitten Floer homology assigned to threemanifold boundaries, (iv) a novel VafaWitten AtiyahFloer correspondence, (v) a proof and generalization of a conjecture by AbouzaidManolescu 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 zerocoupling 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 maximallysupersymmetric topological quantum field theory with electricmagnetic duality in physics.

05262023
Tom A Bridgeland: Geometric structures defined by DonaldsonThomas invariants
Abstract. In recent work the author has explored the idea that the DT invariants of a threedimensional CalabiYau 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 nonlinear 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 nonperturbative 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) Modulitheoretic construction of Joyce structures on spaces of quadratic differentials, (3) Tau function associated to a Joyce structure.

05262023
Changzheng Li：Automorphisms of the quantum cohomology of the Springer resolution and applications
Abstract.In this talk, we will introduce quantum DemazureLusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torusequivariant 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.

05122023
Todor Milanov：Reflection vectors and quantum cohomology of blowups
Abstract: Let X be a smooth projective variety with a semisimple quantum cohomology. The monodromy group of the quantum cohomology is by definition the monodromy group of the socalled 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.

04252023
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.

04182023
Yefeng Shen: Quantum spectrum and Gamma structures for quasihomogeneous polynomials of general type
Abstract. In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of quasihomogenous singularities of general type. Inspired by GalkinGolyshevIritani'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