In complex geometry, the Gauss-Manin connection for a family X over S can be described by the sheaf Omega*_{X/S} of holomorphic de Rham forms. Motivated by mirror symmetry, we look for an A-model analogue using the framework of relative symplectic cohomology developed by U. Varolgunes. Taking a pre-quantum line bundle  L=O(D) over M whose curvature is the symplectic form with fiberwise S^1-action, we consider the S^1-equivariant Hamiltonian Floer theory on the dual L* which comes with a Floer Gysin sequence. We construct a chain homotopy between the Floer-theoretic quantum connection, defined as in the work of P. Seidel and D. Pomerleano, and the connecting homomorphism of the Gysin sequence. Afterwards, we can define a relative Floer Gysin sequence on L*. This is a joint work in progress with C.Y. Mak, D. Pomerleano, and U. Varolgunes.

In this talk, we first review some conjectures of Teleman in his ICM 2014 address and their extensions by Lekili-Segal. Then we highlight how the theory of equivariant Lagrangian correspondence has been used to settle some of them. Based on a joint work (and another work in progress) with S.C. Lau and N.C. Leung, and a joint work in progress with D. Choa, J. Hu, and S.C. Lau.

In this talk, I will report on joint discussions with Kifung Chan and Conan Leung about closed-string mirror symmetry for semi-projective toric varieties, and its extension to the mirror P=W phenomena. This correspondence identifies the perverse filtration associated with the proper affinization map of a semi-projective toric variety with the weight filtration on its mirror.

Let Y be a prequantization bundle over a symplectic manifold M. For a Legendrian submanifold K of Y obtained as a d-to-1 covering of a Lagrangian submanifold L of M, we show that the Rabinowitz Floer homology of K admits a Z_d-action and that its equivariant version is isomorphic to the Floer homology of L. This result can be applied to computations of the Floer homology of L and proofs of non-existences of exact Lagrangian fillings of certain Legendrian submanifolds in Liouville fillings of Y. This is a joint work with Jungsoo Kang and Sungho Kim.

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra, based on the concept of topological–antitopological fusion which was introduced by Cecotti and Vafa. Our main results concern the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. By exploiting a framework introduced by Boalch, we show that this data can be described using Kostant’s theory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classes of regular elements, and it can be visualized on the Coxeter Plane. This is a joint work with Martin Guest.

We discuss examples of symplectic cyclic actions of finite orders that do not extend toHamiltonian circle actions. This is a joint work with Liat Kessler.