Mathematical Sciences: Topology and Its Connections to Geometry and Modular Representation Theory

本课题研究同伦理论及其在几个相关领域中的应用。Friedlander研究了某些代数簇的同伦型，并研究了mod-p李代数的表示理论和李型群的上同调。此外，他还计划继续学习代数K-理论。Mahowald研究稳定同伦和不稳定同伦理论中的内在问题。他的主要重点是球面同伦群中的周期结构。Priddy研究了模表示理论与有限群和无限群的稳定同伦型之间的关系。群的上同调和分类空间的稳定分裂是他工作的中心。简而言之，所有三名研究人员都在使用高度复杂的代数工具来研究几何结构。一方面，它们提高了我们对工具的理解。另一方面，他们应用这些工具来回答关于空间的几何问题。

This project deals with homotopy theory and its applications to several related areas. Friedlander works on the homotopy type of certain algebraic varieties, and also studies the representation theory of mod-p Lie algebras and the cohomology of groups of Lie type. Additionally, he plans to continue his study of algebraic K-theory. Mahowald works on intrinsic questions in stable and unstable homotopy theory. His main emphasis is on periodic structures in the homotopy groups of spheres. Priddy studies relations between modular representation theory and the stable homotopy type of finite and infinite groups. The cohomology of groups and stable splittings of classifying spaces are central to his work. In short, all three investigators are working with highly sophisticated algebraic tools for studying geometric structures. On the one hand, they improve our understanding of the tools. On the other hand, they apply the tools to answer geometric questions about spaces.