기초과학융합연구소

Title: Cluster structures in mirror symmetry.

Abstract: It is expected that a monotone Lagrangian torus together with its disc potential serves an open chart (called a weak Landau-Ginzburg model) of a mirror. In this talk, I will explain a picture that describes how weak LG-models are related to each other in view of cluster theory.

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Title 1 : Noncompact description for Fukaya-Seidel categories of invertible curve singularities

Abstract : For given invertible polynomial W, we consider two types of Fukaya category. One is the usual Fukaya-Seidel category from Lefschetz fibration structure of W. With the maximal symmetry group G of W, we can construct the other Fukaya category of the pair (W,G) from wrapped Fukaya category of the Milnor fiber and quantum cap action of monodromy orbit. In this talk, we compare the equivariant lift of the latter with the Fukaya-Seidel category and prove its derived equivalence for invertible curve singularities. In particular, directedness of the category is obtained from quantum cap action and related constructions. This talk is based on the joint work with Cheol-hyun Cho (SNU) and Dongwook Choa (KIAS) and the other one (in progress) with Hanwool Bae (QSMS), Cheol-hyun Cho (SNU) and Dongwook Choa (KIAS).

Title 2 : Lagrangian Floer theory and quiver representation theory

Abstract : In the mirror symmetry, it is known that some Fukaya type categories of symplectic manifolds are (derived-)equivalent to some quiver representation categories. Using the new type of Fukaya categories introduced in the first talk, we investigate the equivalence for D_n case. In this case, this equivalence gives a geometric way to describe the Auslander-Reiten quiver for D_n quiver representations. This talk is based on the joint work (in progress) with Choel-hyun Cho (SNU).

Title: Symplectic geometry of Gelfand–Zeitlin systems

Abstract: A Gelfand–Zeitlin system is a completely integrable system on a partial flag manifold. In this lecture, I will review the construction of GZ systems and explain how to understand the topology of fibers of GZ systems. Symplectic geometry and mirror symmetry of flag varieties related to GZ systems are also discussed.