UC Berkeley

In this thesis, based on the work completed by the author during his time in graduate school, we explain various ways in which microlocal sheaf methods in symplectic geometry can be used to prove homological mirror symmetry. We explain how tropical methods originally developed by Mikhalkin can be used to prove homological mirror symmetry for any hypersurface in (C*)^n, and we also present a proof of homological mirror symmetry in the case of multiplicative hypertoric varieties, emphasizing the features which we expect will prove common to all K-theoretic Coulomb branches.