Applications of symplectic geometry

辛几何的应用

• 批准号: 170264-2011
• 负责人: Jeffrey, Lisa
• 金额: $ 3.06万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569564
• 关键词: Applications; symplectic; geometry

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).

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。 第二个主题是内爆横截面。如果辛流形承认非阿贝尔李群的哈密顿作用（例如 SU(2)），则可以采用最大李代数的原像 环面。辛内爆的目的是使空间塌缩，以便我们通过最大环面的哈密顿作用恢复辛空间。 第三个主题是 hyperkaehler 商。 超凯勒结构由辛流形上的三个可交换的近复结构组成，它们都与辛结构兼容，并通过四元数相互关联。 Hyperkaehler 流形是维度可被 4 整除的辛流形，并且几乎总是非紧的。 如果超凯勒流形承认李群保持辛结构的作用，则可以形成超凯勒商，这是一种将维度减少 4 的运算。迫切感兴趣的问题是从超凯勒流形的等变上同调到商的普通上同调的自然映射是否是满射。 第四个主题是基础循环组。环群是从圆到群 G 的映射集。基环群是将圆中的基点映射到 G 中的单位元的映射集。它是一个辛流形，并承认 G 的最大环面的哈密顿作用。Atiyah 和 Pressley 证明了对此的凸性的类似。我和我的合作者表明，该动作的矩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) 时基环群的等变 K 理论的环结构。