Symplectic Geometry and Stratified Spaces

Reyer Sjamaar利用微分拓扑学、奇点理论和不变量理论的方法研究了辛分层空间和哈密顿李群作用的不变量。该项目有望有助于理解由哈密顿李群作用和动量图引起的奇点。其中一个目标是计算辛商和辛截面的等变指标。进一步的目标是构造分层辛空间的同调Todd类，并将其应用于具有奇点的流形的指标理论。辛几何是几何学的一个分支，它支配着牛顿经典力学的定律。经典力学是对物理世界中物体大尺度行为的一种相当精确的描述。在亚微观尺度上，量子力学定律给出了更精确的描述。为了将宏观和微观行为联系起来，能够在这两种描述之间来回转换是很重要的，这在本世纪引起了物理学家和数学家的大量活动。我的项目旨在为那些表现出“坏”或奇异特征的系统的量化做出贡献，这些特征无法用传统方法处理。这种奇点通常出现在线性、球形或更复杂的对称性中。物理对象系统中的对称性通常使人们能够预测其未来行为的重要特征，而我的目标是找出这些特征如何在量子水平上反映出来。

Reyer Sjamaar proposes to investigate invariants of symplectic stratified spaces and of Hamiltonian Lie group actions, using methods from differential topology, singularity theory and invariant theory. This project is expected to contribute to the understanding of the singularities that arise from Hamiltonian Lie group actions and momentum maps. One of the goals is to calculate the equivariant index of symplectic quotients and symplectic cross-sections. A further goal is to construct homology Todd classes for stratified symplectic spaces, with applications to the index theory of manifolds with singularities. Symplectic geometry is the branch of geometry governing the laws of the classical mechanics of Newton. Classical mechanics is a quite accurate description of the large-scale behaviour of objects in the physical world. At submicroscopic scales a more precise description is given by the laws of quantum mechanics. To link together macroscopic and microscopic behaviour, it is of importance to be able to go back and forth between the two descriptions, and this has generated much activity by physicists and mathematicians in this century. My project intends to contribute to the quantization of systems that exhibit ``bad'', or singular, features that cannot be treated by conventional means. Such singularities often arise in the presence of linear, spherical, or more complicated types of symmetry. Symmetry in a system of physical objects often enables one to predict important features of its future behaviour, and my object is to find out how these features are reflected at the quantum level.