UC Berkeley

The main goal of this dissertation is to import symplectic geometric methods into microlocal sheaf theory based on the foundational sheaf quantization construction by Guillermou, Kashiwara, and Schapira. This construction provides the notion of isotopies of sheaves and a sheaf-theoretic analogue of the notion of continuation maps in Lagrangian Floer theory.

Based on previous work by Ganatra, Pardon, and Shende, we make further ex- amination on the category of unbounded sheaves microsupported in some singular isotropic Λ in the cosphere bundle. We show that various categorical constructions concerning this category can be described in symplectic geometric terms by using isotopies of sheaves.

The main construction is a sheaf-theoretic analogue of the wrapped Fukaya category, by localizing a category of sheaves microsupported away from some given Λ along continuation maps. When Λ is a subanalytic singular isotropic, we also construct a comparison map to the category of compact objects in the category mentioned above, and show that it is an equivalence. The last statement can be seen as a sheaf-theoretical incarnation of the sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende.