This course will give introductions to research-related topics in mathematics. The topics will vary from semester to semester.

This course will focus on recent applications of the microlocal theory of sheaves to symplectic topology and homological mirror symmetry. We begin with a quick account of the microlocal theory of sheaves, after Kashiwara and Schapira, explaining the main structures and properties but not spending much time on the various technical foundations and difficult proofs. Next will be some applications of this theory to symplectic topology (nearby Lagrangians, capacities, categorical invariants) and homological mirror symmetry (mirrors of toric varieties, cluster structures from Legendrian knots). Finally we will study the recent generalization of microsheaf theory to arbitrary contact / exact symplectic manifolds, and corresponding results, (homological mirror symmetry for very affine hypersurfaces, perverse t-structures, microlocal Riemann-Hilbert). Open questions will be presented throughout.

Prerequisites: Familiarity with homological algebra and the language of sheaves, algebraic geometry, symplectic geometry.

The first meeting will be on Thursday 1/19.

Repeat Rules