Derived Categories and Mirror Symmetry

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Mirror symmetry began as a duality between superconformal field theories. In 1991, Maxim Kontsevich re-interpreted this concept from physics as an incredibly deep and ubiquitous mathematical duality now known as homological mirror symmetry (HMS). In his famous Fields Medal address, he proposed a new type of geometry based on derived categories, creating a frenzy of activity in the mathematical community which lead to a remarkable synergy of diverse mathematical disciplines: symplectic geometry, algebraic geometry, and category theory. HMS is now the cornerstone of an immense field of active mathematical research which utilizes ever-advancing state-of-the-art techniques from string theory.***This proposal constitutes a 5-year plan to study derived categories in algebraic geometry and more specifically, their relationship to constructions in string theory and mirror symmetry.  Based on my previous work, I expect applications to range from partial solutions of the Hodge Conjecture (a "Millennium Problem")  to a deep conjecture of Kawamata on the relationship between birational geometry and derived categories.  I intend to fully solve the latter problem.  In doing so, I will be aided by an international network of collaborators which includes Kontsevich himself.  ***The central mathematical objectives of this proposal are:***1) Study algebraic varieties with group actions via stratifications with the intent to use these stratifications to extend the relationship between geometric invariant theory and derived categories.***2) Unify, develop, and expand mirror symmetry constructions with gauge groups in the setting of derived categories and apply the theory above towards classical algebro-geometric invariants: Noether-Lefschetz loci, algebraic cycles (and the Hodge conjecture), Griffiths groups.***The first objective is in many ways a fundamental study of group actions in algebraic geometry.  On the other hand, it has enormous potential for furthering the specific interests of my field.  Indeed, through Objective 1, I envision a new understanding of semi-orthogonal decompositions and equivalences of derived categories which I expect to re-write and unify the current literature.  This is highly evidenced by my previous work on derived categories and geometric invariant theory which recovers many of the most exciting theorems in derived categories and algebraic geometry.***The second objective has an outward focus on the greater field of mirror symmetry and algebraic geometry.  Much of my previous work has been building towards this objective. Developing mirror symmetry constructions with gauge groups is natural to the existing literature and is of great importance, as it expands on the highly influential role of toric mirror symmetry.  In addition, applications towards classical algebro-geometric invariants are of great interest to the larger algebraic geometry community. *** *** *** *** **