The topology of sympletic and hyperkahler quotients

辛几何是经典物理学的自然数学框架。经典力学系统的相空间，参数化粒子的位置和动量，可以被解释为辛流形。其他重要的例子出现在物理学中，特别是弦论中，我的工作包括理解这类系统的对称性。对称性的存在，除了辛几何结构，帮助我们在计算系统的拓扑和几何不变量，并允许我们执行显式calculations.My拟议的研究扩展目前已知的方法辛几何，特别是那些显式计算这些系统的拓扑性质的方法。这些前沿的进展将有可能应用于数学和物理学的许多领域，如表示论（量子物理学的数学框架）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.