After briefly reviewing the SYZ mirror construction of log Calabi-Yau surfaces, we introduce a tropical geometric method for locating critical points of Landau-Ginzburg potentials. We investigate the change in the number/location of critical points of the potential under blowups and blowdowns.

I will explain the content of Haussmann and Knutson's paper and provide the necessary background for a project joint with Jinju Choi, describing the topology of the residual completely integrable systems of Gelfand–Zeitlin systems on weight varieties. 

We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkaehler fourfolds among them. This implies that a hyperkaehler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville-Donagi and Debarre-Voisin. This is a joint work with Eunjeong Lee at Chungbuk National University.

Polygon space is a moduli space of closed linkage in Euclidean spaces. The length map on this space encodes the side lengths of polygons. For Euclidean plane, the connectivity of the fibers of length map has been studied with respect to the chamber structures by Kapovich and Millson(1995). The image of the length map is a convex polytope-a rectified simplex-obtained by truncating of a regular simplex. In this talk, we explore the chamber structures of the image of the length map, with a focus on its relationship with the rectified simplex and the associated polygons.

I’ll review several constructions of a Landau-Ginzburg mirror for a punctured Riemann surface, and discuss how to achieve its closed-string 

mirror symmetry, mainly focusing on the noncommutative mirror introduced by Bocklandt. Our approach involves a geometrically constructed ring homomorphism from the symplectic cohomology of the surface to some Hochschild-type invariant of the mirror matrix factorization category. 

When a torus acts on a symplectic manifold in a Hamiltonian way, we are given a moment map whose image is a convex polytope. One can recover the symplectic manifold from the polytope if the dimension of the torus is maximal. The manifold is constructed as the quotient space of an open subset of complex Euclidean space and carries both symplectic and complex structures. We will review this construction and see the relation to the quotient by algebraic torus.