Topology of Symplectic Algebraic Varieties

辛代数簇的某些特殊类在数学的各个领域中起着中心作用。 例如，超环面簇的几何和拓扑性质为超平面排列的拓扑和拟阵的组合学提供了新的线索。 在表示论中，李代数簇提供了无限维李代数作用的几何实现，导致了规范基，有时导致了范畴上的作用。 非交换霍奇理论是对三个相关辛簇的研究，最近被研究为朗兰兹对偶和镜像对称之间的联系。 这个项目将独立地为每一项工作做出贡献，同时也将推进一种统一我们对各种个别现象的理解的共同处理。辛代数簇是在组合学和表示论的数学领域中自然出现的几何对象，由于它们在弦理论中的存在，也引起了物理学家的兴趣。 在某些情况下，涉及上述领域之一的这些空间的某些想法可以转移到另一个领域，并产生令人惊讶的结果。 以这种方式，研究者计划研究辛代数簇在其各种化身中的一般和具体特征。

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.