Hilbert Schemes of Plane Curve Singularities and Matrix Factorizations

Explore the connections between algebraic geometry and low dimensional topology in this lecture on Hilbert schemes of plane curve singularities and matrix factorizations. Delve into the work of Oblomkov-Shende and Maulik, which demonstrates how the HOMFLY-PT polynomial of algebraic links can be expressed using Hilbert schemes. Examine the growing interest in mirror symmetry of hypersurface singularities, with a focus on plane curve singularities as natural testing grounds for the mirror symmetry conjecture. Investigate the relationships between Hilbert schemes of plane curve singularities, topological data of algebraic links, and matrix factorizations, including stability conditions on these structures.