Applications of symplectic geometry

我的大部分研究都集中在2-流形的基本群的表示的共轭类的模空间上。 第二个话题是内爆横截面。如果辛流形允许非阿贝尔李群(例如SU(2))的哈密顿作用，则可以取极大李代数的原象 托鲁斯。辛内爆的目的是使空间崩塌，从而恢复一个具有极大环面的哈密顿作用的辛空间。 第三个话题是超凯勒商。超Kaehler结构由辛流形上的三个可交换的近复结构组成，它们都与辛结构相容，并通过四元数相互联系。Hyperkaehler流形是维度可被4整除的辛流形，并且几乎总是非紧的。如果Hyperkaehler流形允许保持辛结构的李群的作用，则可以形成Hyperkaehler商，这是一个降维4的运算。一个迫切感兴趣的问题是，从Hyperkaehler流形的等变上同调到商的普通上同调的自然映射是否满射。 第四个主题是基本循环组。循环群是从圆到群G的映射的集合，基于循环群是将圆中的基点映射到G中的单位元的映射的集合。它是一个辛流形，并且允许G.Atiyah的极大环面的哈密顿作用，Pressley证明了它的一个类似的凸性。我和我的合作者证明了这一行动的时刻地图的水平集是相连的。在第二个项目中，我和我的合作者(Harada和Selick)研究了当G=SU(2)时基础环群的等变K-理论的环结构。

Much of my research has concentrated on the moduli space of conjugacy classes of representations of the fundamental group of a 2-manifold. A second topic is imploded cross-sections. If a symplectic manifold admits the Hamiltonian action of a nonabelian Lie group (for example SU(2)), one may take the preimage of the Lie algebra of the maximal torus. The purpose of symplectic implosion is to collapse the space so that we recover a symplectic space with a Hamiltonian action of the maximal torus. A third topic is hyperkaehler quotients. A hyperkaehler structure consists of three commuting almost complex structures on a symplectic manifold, which are all compatible with the symplectic structure and are related to each other via the quaternions. Hyperkaehler manifolds are symplectic manifolds of dimension divisible by 4, and are almost always noncompact. If a hyperkaehler manifold admits the action of a Lie group preserving the symplectic structure, one may form the hyperkaehler quotient, an operation which reduces the dimension by 4. A question of urgent interest is whether the natural map from the equivariant cohomology of a hyperkaehler manifold to the ordinary cohomology of the quotient is surjective. A fourth topic is the based loop group. The loop group is the set of maps from the circle to a group G. The based loop group is the set of maps which map the basepoint in the circle to the identity element in G. It is a symplectic manifold, and admits a Hamiltonian action of the maximal torus of G. Atiyah and Pressley proved an analogue of convexity for this. I and my collaborators showed that the level sets of the moment map for this action are connected. In a second project, I and my collaborators (Harada and Selick) have studied the ring structure of the equivariant K-theory of the based loop group whenG=SU(2).