The topology of sympletic and hyperkahler quotients

辛几何是经典物理学的自然数学框架。一个经典力学系统的相空间，既参数化了粒子的位置又参数化了粒子的动量，它可以被解释为辛流形。其他重要的例子出现在物理学中，特别是在弦理论中。我的工作包括理解这些系统的对称性。对称性的存在，加上辛几何结构，帮助我们计算系统的拓扑和几何不变量，并允许我们执行显式计算。我提出的研究方法是扩展目前已知的辛几何方法，特别是这些系统的拓扑性质的显式计算。这些前沿的进展将有可能应用于数学和物理的许多领域，如表示理论（量子物理的数学框架）和弦理论。

Symplectic geometry is the natural mathematical framework of classical physics. The phase space of a classical mechanical system, which parametrizes both the position and the momentum of particles, can be interpreted as a symplectic manifold. Other important examples arise in physics, especially in string theory.My work involves understanding the symmetries of such systems. The presence of the symmetry, in addition to the symplectic-geomtric structure, aids us in computing topological and geometric invariants of the system, and allow us to perform explicit computations.My proposed research studies ways of extending the currently known methods of symplectic geometry, in particular those for the explicit computation of topological properties of these systems. Progress on these fronts will have possible applications to many areas of mathematics and physics, such as representation theory (the mathematical framework of quantum physics) and string theory.