Interesting and impactful mathematics often arises when new connections are made between different fields of math. While even heuristic connections can be fruitful, mirror symmetry provides a fascinating direct connection, originating from modern physics, between algebraic geometry and symplectic topology that has led to major advances in both areas. Algebraic geometry is a rich and classical field of mathematics that explores shapes called algebraic varieties described by polynomial equations. Symplectic topology is a younger area that studies shapes built from a geometric formalism for classical mechanics by packaging solutions to certain partial differential equations into algebraic invariants. This project aims to deepen our understanding of the mirror symmetry phenomenon by building on new insights in a special case where the algebraic varieties are particularly symmetric. This will be done with the aim of verifying new cases of the homological mirror symmetry conjecture, exploring structural aspects of a symplectic invariant known as the Fukaya category, and investigating arithmetic aspects of mirror symmetry. The project will also involve undergraduate research projects on combinatorial problems coming from mirror symmetry. The first technical goal of the project is to further develop functorial aspects of the toric homological mirror symmetry equivalence by enlarging the list of sheaves and functors that can be provably described in terms of Lagrangian submanifolds