The goal of this project is to expand our understanding of mirror symmetry and illustrate how it incorporates new dualities in mathematics. In particular, the homological mirror symmetry conjecture of Kontsevich expresses this physical phenomenon as an equivalence of brane categories. One such category is the Fukaya category, constructed from Lagrangian submanifolds. A central focus of this project will be the relationship, found recently by the Principal Investigator with D. Nadler, between the Fukaya category of the cotangent bundle of a manifold and constructible sheaves on the base. This construction will be used: 1) to study mirror symmetry for the cotangent bundle; 2) to characterize perverse sheaves in terms of their corresponding Fukaya objects; and 3) to construct Hecke eigensheaves in the geometric Langlands program, following the proposal of Kapustin-Witten to look at torus fibers of the Hitchin fibration. Another aspect of the proposal focusses on constructing the mirror map of moduli purely from techniques of homological mirror symmetry, thereby bridging Kontsevich's point of view with the historical approach to mirror symmetry.
Over the past few decades, mathematics and theoretical physics have become linked in a profound way. The field of string theory best illustrates this interdependence. Within string theory, mirror symmetry is the prime example of how one phenomenon in physics can link seemingly disparate fields of mathematics. This proposal will broaden the mathematical scope of mirror symmetry through lines of research that both create, and capitalize on, recent advances. In particular, connections between topology, representation theory, and the geometric Langlands program have emerged, as have new methods for constructing examples of mirror symmetry from homological algebra. This research could thus lead to progress in several longstanding goals: understanding mirror symmetry in its most general setting, and constructing eigensheaves for Hecke operations in the geometric Langlands program.
