This project is an attempt to apply new cohomological and representation-theoretic ideas to problems in transformation groups and homotopy theory. The particular aspects which it involves can be described as follows: (1) Incorporate the study of certain discrete groups, such as arithmetic groups and mapping class groups, to an existing program based on applying methods from modular representation theory to group actions. (2) Determine general properties of the cohomology of finite groups, and how, using equivariant cohomology, this influences the role they play as transformation groups. (3) Obtain necessary and sufficient conditions for a finite group to act freely on a product of spheres, and analyze the restrictions this imposes on the homology representations when they are of the same dimension. This work is aimed at approaching a global conjecture on the free rank of symmetry of a finite complex. (4) Use theory of infinite loop spaces and invariant theory to obtain precise cohomological information on the symmetric groups, with several applications in mind. They include describing the cohomology of certain double covers of the symmetric groups which serve as finite models for a space which detects important characteristic classes. Other applications to stable homotopy and group cohomology are also planned. The invariant theory used is seen as an interesting and important algebraic problem in its own right, encoding most of the cohomological structure of the symmetric groups. In other words, the various parts of this project all concern the symmetry of geometric objects. Algebraic tools for making calculations about symmetry will be perfected, and these tools will then be applied to answer natural questions.
