Certain special classes of symplectic algebraic varieties play central roles in various areas of mathematics. For example, geometric and topological properties of hypertoric varieties have shed new light on the topology of hyperplane arrangements and the combinatorics of matroids. In representation theory, quiver varieties provide geometric realizations of actions of infinite dimensional Lie algebras, leading to canonical bases and to sometimes to actions on categories. Nonabelian Hodge theory is the study of three related symplectic varieties, and has recently been investigated as a link between Langlands duality and mirror symmetry. This project will contribute to each of these endeavors independently, and will also advance a common treatment that unifies our understanding of the various individual phenomena.
Symplectic algebraic varieties are geometric objects that arise naturally in the mathematical fields of combinatorics and representation theory, and are also of interest to physicists due to their presence in string theory. In some instances, certain ideas involving these spaces in one of the above mentioned fields can be transported to another one with surprising results. In this manner, the investigator plans to study both the general and the specific characteristics of symplectic algebraic varieties in their various incarnations.
