Mathematical Sciences: Transformation Groups on Manifolds and Varieties

本课题研究紧李群在流形上的光滑作用和代数群在仿射变异上的代数作用。这里有两个代数问题：有限（约化）群在非奇异变量V上的代数作用的共轭类是否可数？如果V是复n空间，这个数是有限的吗？在光滑设置下，项目的目标之一是：用不动点数据给出非自由情况下（有利情况下）等变法线映射的等变手术障碍信息，并提供保证这些障碍消失的条件（即给出消失定理）。群体行为在自然界中无处不在。理解它们类似于理解动力系统。

This project deals with smooth actions of compact Lie groups on manifolds and algebraic actions of algebraic groups on affine varieties. Here are two of the many problems in the algebraic setting: Are conjugacy classes of algebraic actions of a finite (reductive) group on a non-singular variety V countable? In the case V is complex n-space, is this number finite? Among the aims of the project in the smooth setting: Give information on the equivariant surgery obstructions for an equivariant normal map in the non-free case (in favorable cases) in terms of fixed point data, and provide conditions which guarantee these obstructions vanish (i.e. give vanishing theorems). Group actions are ubiquitous in nature. Understanding them is akin to understanding dynamical systems.