Homological mirror symmetry (HMS) is a research program proposed by Kontsevich in 1994 ICM. It provides an unexpected connection between symplectic geometry (A-side) and algebraic geometry (B-side), and has also recently been found to be related to representation theory.
In this talk, we focus on the mirror symmetry "Degenerate cusp singularities (B-side) <-> Riemann surfaces (A-side)" and their representations. First we review the representation theory of Cohen-Macaulay modules over degenerate cusps, and introduce degenerate vector bundles to give its geometric meaning (B-side). Then we explain how they correspond to holonomy of geodesics in the mirror Riemann surface (A-side) under HMS.
This talk is based on a joint work with Cheol-Hyun Cho, Wonbo Jeong and Kyoungmo Kim.