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According to the Kempf–Ness theorem, the GIT quotient is equivalent to the symplectic reduction. Using this correspondence, without equivariant Floer theory, we explain how to relate the counting of holomorphic disks between a projective variety equipped with a linear algebraic group action and its symplectic reduction.

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Mirror symmetry predicts a deep relation between moduli of Lagrangian branes and coherent sheaves. On the algebraic side, locally, moduli stack of coherent sheaves are often described by quiver varieties via deformation theory. From the symplectic perspective, Floer theory plays the role of Ext, and it is natural to expect a quiver description of moduli of Lagrangian branes.

In this talk, I will recall recent results realizing Nakajima quiver varieties as deformation spaces of framed Lagrangian branes. I will then focus on bulk deformations and explain how they correspond to noncommutative deformations of the mirror quiver algebra. This yields a symplectic framework for noncommutative ADHM-type constructions and the noncommutative models introduced by Kawamata. Finally, I will explain how these results fit into local homological mirror symmetry. This talk is based on an ongoing project with Hansol Hong and Siu-Cheong Lau.

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Any meromorphic connection on the tangent bundle of a Riemann surface defines, away from its poles, a complex affine structure on the surface. Building on this geometric interpretation, we provide the formal and analytic classifications of these poles, as well as their local geometric models. As an application, we show how the Delaunay triangulation of a set of points in the plane admits a broad generalization, yielding a canonical decomposition of any affine surface arising from a meromorphic connection. This is a work in collaboration with Xavier Buff and Arnaud Chéritat. 

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In this talk, I will first briefly overview the geometry of the moduli space of parabolic Higgs bundles.Then I will discuss the spectral correspondence, which produces examples of symplectic resolutions equipped with structures of algebraic integrable systems. This is ongoing joint work with Jiachoon Lee.

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In this talk, we discuss a construction of Iwahori-Coulomb branch action on the equivariant quantum cohomology of a smooth semiprojective variety. Applying to the case of cotangent bundles of flag varieties, this yields a trigonometric DAHA action which can be explicitly computed in terms of Demazure-Lusztig operators and stable envelopes. As applications, we explain: (i) how this recovers the Peterson homomorphism by taking the confluent limit; (ii) a proof of the conjecture that, up to a shift of the dilation parameter, the original BFN Coulomb branch for the coadjoint matters is isomorphic to the spherical trigonometric DAHA.

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The homology of a compact Lie group is a Hopf algebra, and in particular a bialgebra. I will present a chain-level counterpart of bialgebras that forms the objects of an infinity category, and that permits to describe compact Lie group actions on Morse complexes and Fukaya categories, in a functorial way. This is related to instanton gauge theory and extended topological field theory. This will be based on arXiv:2404.17393, arXiv:2410.16225 with A. Hock and T. Mazuir, and  arXiv:2505.01362.

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The mirror symmetric Gamma conjecture states the equality between the central charges of a pair of mirror objects under homological mirror symmetry. I will show that for a toric Fano orbifold and its Landau-Ginzburg mirror, it naturally arises as the Fourier transform of the equivariant version of mirror symmetric Gamma conjecture on C^d. I will then present an interpretation of this result within the framework of 3D mirror symmetry. This is based on joint work with Konstantin Aleshkin and Bohan Fang, as well as ongoing joint work with Conan Leung.

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Brane quantization was introduced by Gukov-Witten to quantize symplectic manifold. They predicted that the geometric quantization of the symplectic manifold admits an action from a certain (holomorphic) deformation quantization of its complexification. In this talk, I will present a mathematical formulation of Gukov-Witten proposal vis SYZ mirror symmetry. We begin with a semi-flat SYZ fibration that admits a space filling coisotropic A-brane and use family geometric quantization to construct its mirror E. We then use family Toeplitz construction (with a twist) to obtain an ""asymptotically faithful"" representation of the Moyal star product on the coisotropic A-brane. The Gokov-Witten proposal in this case can simply be understood as an action of End(E) on a fiber of E. This is a joint work with Kwokwai Chan, Conan Leung, Qin Li, and Tony Yau.

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Seidel representation S is an important invariant which provides interesting relationships between enumerative geometry, Hamiltonian diffeomorphism group and (2d or 3d) mirror symmetry. For a Hamiltonian S^1 space X, S is obtained from counting pseudoholomorphic sections of its Seidel space E.

In this talk, we will discuss ongoing joint work with Lau and Leung on resolving a conjecture of Teleman on identifying S with Teleman's mirror fibration $F$ under closed-string mirror symmetry of X, which generalises a result of Chan-Lau-Leung-Tseng when X is toric semi-Fano. A key ingredient of the proof is the equivariant Floer theory of the quotient correspondence between E and X as a symplectic S^1-quotient of E, based on an earlier joint work with the same authors. In particular, we will show its weak unobstructedness under bulk deformation by S.

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It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory. It is a joint work with Wonbo Jeong and Beom-Seok Kim.