Cohomology and Actions of Finite Groups

DMS-9987434 Alejandro Adem.在这个项目中，主要研究人员建议使用代数和拓扑方法来研究变换群、群上同调及相关主题中的一些基本问题。特别是，他建议通过群作用来研究群上同调中的周期性现象；分析由场论中的基本问题引起的某些伽罗瓦群的上同调；计算低秩散发单群的上同调；以及研究一些代数和拓扑明显相互作用的杂项问题。拓扑学测量数学对象在扭曲和拉动等变形下不变的内在性质。特别是，用拓扑学方法研究对称性是卓有成效的。例子来自物理学和其他自然科学，为他们的研究获得精确的数学方法是一个具有挑战性的课题。该项目建议使用最近发展的数学技术以及计算机辅助计算来实现这一点，在一些已知具有重大意义的不同情况下。

DMS-9987434Alejandro AdemIn this project the principal investigator proposes to use both algebraic and topological methods to study a number of basic questions in transformation groups, group cohomology and related topics. In particular he proposes to study the phenomenon of periodicity in group cohomology via group actions; to analyze the cohomology of certain Galois groups arising from basic questions in field theory; to calculate the cohomology of low rank sporadic simple groups; and to investigate a number of miscellaneous problems where a significant interplay of algebra and topology is apparent.Topology measures intrinsic properties of mathematical objects which are invariant under deformations like twisting and pulling. In particular symmetries can be fruitfully studied by using topological methods. Examples arise from physics as well as other natural sciences and obtaining precise mathematical methods for their study is a challenging topic. This project proposes to use recently developed mathematical techniques as well as computer assisted calculations to achieve this in anumber of different instances which are known to be of significant interest.