Explore the intricate connections between mirror symmetry and toric varieties in this 52-minute conference talk. Delve into how mirror symmetry predicts an action by the fundamental group of a stringy Kähler moduli space on the derived category of an algebraic variety, with particular focus on toric varieties where models for this space are understood but constructing the action remains an open problem. Examine the specific case of the A_{n-1} singularity, where the action can be studied through a moduli space of Legendrians isotopic to the FLTZ Legendrian, ultimately recovering a braid group action first discovered by Seidel and Thomas. Learn about cutting-edge research in singularity theory that bridges algebraic geometry, symplectic topology, and mirror symmetry, presented as part of the "New Developments in Singularity Theory" conference. Gain insights into collaborative work with Michela Barbieri and Jeff Hicks that advances our understanding of topological aspects of mirror symmetry in the toric setting.