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.
