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Koszul duality is a classical correspondence between an algebra and a dual algebra, with natural manifestations in topology and geometry. In this talk, I will briefly review classical Koszul duality, and then explain how a similar picture arises in local mirror symmetry. I will also explain how the extended localized mirror construction, introduced by Cho–Hong–Lau, can be understood in terms of Koszul duality, and how this perspective can be used to compute compact Fukaya categories and recover some known generation results. This talk is based on joint work with Hansol Hong and Siu-Cheong Lau.

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Given a Hamiltonian G-space Y with a moment map, its moment Lagrangian L is a Lagrangian correspondence from T*G to Y x Y which serves as the action in Weinstein's symplectic category formalism. Since Teleman’s proposal on its role in topological action on the Fukaya category Fuk(Y) of Y, its applications to symplectic topology emerged, in particular the generation results of Fuk(Y) in the monotone setting due to Evans and Lekili.

In this talk, I will prove the tautological unobstructedness of L in a fairly general setting. This provides an initial step towards the study of Lagrangian Floer theory of L beyond monotone setting. The key idea is to realize L as the fixed locus of an anti-symplectic involution, which turns out to be the inversion of a new symplectic groupoid structure on T*G x Y x Y, another result which might be of independent interest. This talk is based on my work available at arXiv:2609.12033.”

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: In perturbative quantum field theory, the divergence of Feynman graph integrals is one of the central problems. In this talk, we prove that such divergences do not occur in holomorphic quantum field theories on closed real-analytic Kähler manifolds. Our proof is based on the theory of wonderful compactifications in algebraic geometry, which provides a geometric understanding of these integrals. As a consequence, we establish a gauge anomaly formula for these graph integrals. This is joint work with Junrong Yan.

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We present a complete classification of the formality of configuration spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$. For $(n'=0,n\ge2)$ or $(n'=1,n\ge1)$, the configuration spaces are non-formal. For $n'\ge2,n\ge1$, we establish an explicit quasi-isomorphism between the CDGA of differential forms on configuration spaces and its cohomology by using a diagram complex and a regularized configuration space integral, which leads to the formality. In the formal case, we get a higher dimensional analog of vertex algebras. The talk is based on arXiv:2608.28200.

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For finite-dimensional exponential integrals, Picard–Lefschetz theory relates critical points, Lefschetz thimbles and Stokes phenomena, while resurgence connects differences of critical values with Borel singularities. This provides a natural model for studying heat kernels from an infinite-dimensional viewpoint.

In this talk, I will discuss how this picture extends to the complexified path space of a real-analytic Riemannian manifold. Geodesics play the role of critical points of the energy functional, and holomorphic geodesics give rise to complex WKB sectors. This suggests a correspondence between complex geodesic energies, Borel singularities of the heat-kernel expansion, and Picard–Lefschetz trajectories. As a guiding example, I will discuss the hyperbolic plane, where the complex quadric model gives an explicit framework for studying complexified paths, Morse flow, Stokes phases and their expected relation to the resurgence structure of heat-kernel asymptotics. This is a joint work with Si Li and Yong Li.

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Following the work of Beilinson–Drinfeld, Tamarkin, and Costello–Gwilliam, I will introduce OPE algebras on Riemann surfaces and, more generally, on smooth complex varieties. This framework connects the physical intuition of operator product expansions with the modern theories of chiral and factorization algebras. I will then discuss infinite-dimensional symmetry algebras arising in this setting, including higher Kac–Moody algebras, as studied by Faonte, Hennion, and Kapranov and by Gwilliam and Williams, as well as higher Virasoro algebras. For the latter, I will describe recent joint work with Brian Williams, together with related computations of central extensions carried out with Minghao Wang and Brian Williams. I will also explain how the central charges of free field theories are governed by the local family index theorem. I will conclude with a brief overview of related developments and several directions for future research.

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The Donaldson–Fujiki moment map picture interprets the scalar curvature of a Kähler metric as a moment map for the action of Hamiltonian diffeomorphisms on the space of compatible complex structures. We generalize this picture to other scalar curvature quantities using equivariant determinant line bundles. Interpreting the complexified orbit as varying Kähler potentials while keeping the complex structure fixed, these moment maps arise as variations of the log of the Quillen metric on determinant line bundles. This framework also naturally yields other formal GIT counterparts, such as log-norm functionals (~Mabuchi functional) and weights (~Futaki invariant).

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Quantum cluster algebra was introduced by Berenstein and Zelevinsky. As its name suggests, it is a quantization of the classical cluster algebra. Cluster algebras always come in pairs, the cluster ensemble $(\mathcal{A},\mathcal{X})$. It is believed that they admit a mirror partner $(\mathcal{A}^{\vee},\mathcal{X}^{\vee})$, which is also a cluster ensemble. In this talk, we show that, after specialization, the $\mathcal{A}^{\vee}$-quantum cluster algebra can be viewed as the Hom-space of a canonical coisotropic A-brane $\mathcal{B}_{cc}$ on $\mathcal{A}^{\vee}$ by using SYZ mirror symmetry. More precisely, we use family geometric quantization (with real polarization) to define the mirror B-brane $\check{\mathcal{B}}_{cc}$ and prove that for each quantization level $k$, $Hom_B(\check{\mathcal{B}}_{cc}^{(k)},\check{\mathcal{B}}_{cc}^{(k)})$ is isomorphic to the quantum cluster algebra. This is a joint work with Hengzhi Chin and Yu-Tung Tony Yau.

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We introduce flippered hyperbolic surfaces and study the renormalized volumes of their moduli spaces. These surfaces may be viewed as natural generalizations of the hyperbolic uniformizations of oriented open-string worldsheets. We present three geometric recursions for these volumes, with particular emphasis on flippered disks and annuli.