Online Seminar
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09222023
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.
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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 A-type cluster algebra on quantum cohomology of flag variety.This is a joint work with Weiqiang He.
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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 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.
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08092023
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.
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05262023
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.
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05262023
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.
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05122023
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.
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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.
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04182023
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
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04102023
Cheng Shu: Mirror of Orbifold Singularities in the Hitchin Fibration
Abstract. For the group SLn, we study the geometry of singular Hitchin fibres over the elliptic locus of the Hitchin base. This gives interesting information on the Fourier-Mukai transform of a skyscraper sheaf supported at an orbifold singularity of the Hitchin moduli space for PGLn. The main results prove a classical version of the conjectures of Frenkel-Witten concerning the mirror of orbifold singularities.
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03282023
Longting Wu: All-genus WDVV recursion, quivers, and BPS invariants
Abstract: Let D be a smooth rational ample divisor in a smooth projective surface X. In this talk, we will present a simple uniform recursive formula for (primary) Gromov-Witten invariants of O_X(-D). The recursive formula can be used to determine such invariants for all genera once some initial data is known. The proof relies on a correspondence between all-genus Gromov–Witten invariants and refined Donaldson–Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations. This is a joint work with Pierrick Bousseau.
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03232023
Abstract. Assuming certain comparison between non-commutative Hodge structures with classical Hodge structures, we prove the CEI associated with a smooth projective family of Calabi-Yau's satisfy the holomorphic anomaly equation. This is based on a work in progress with Yefeng Shen.
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03232023
Junwu Tu: An introduction to categorical enumerative invariants (CEI)
Abstract. In this talk, we shall present the definition of CEI associated with smooth and proper Calabi-Yau categories. We also sketch an explicit combinatorial formula of CEI. In the end, we discuss about concrete computations as well as some interesting questions. The talk is based on joint works with Andrei Caldararu and Lino Amorim.
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03172023
Yalong Cao: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.