Cohomology and Actions of Finite Groups

首席研究员将运用代数和拓扑学的技术来研究变换群、轨道、场论、群上同调和相关主题中的一些基本问题。其中，他提出了以下研究项目：(1)利用群上同调和同伦理论的方法构造有限群作用；(2)辫状群和其他离散群的表示空间和k理论研究；(3)轨道的代数拓扑：轨道相关拓扑不变量的发展与研究；(4)群的场论与上同调：用上同调方法分析伽罗瓦群；(5)采用理论与计算机辅助相结合的方法计算了低秩单群的上同调。这个提议在概念上是跨学科的，因为它打算结合代数和拓扑的方法来解决广泛的问题。对称性是数学中的一个中心话题，可以使用拓扑方法有效地研究对称性。对轨道弦理论数学方面的研究是几何与物理融合的代表。许多重要的思想起源于物理学，它们不断地导致数学新领域的开辟。另一方面，群的上同调是一个涉及复杂代数不变量的课题，它在表示论、数论、场论等领域中都很自然地发挥着作用。最近的技术使用了计算机代数的先进方法和能力，所提出的计算将成为该领域进一步计算和理论发展的基准，具有许多潜在的应用。本建议将寻求在拓扑学、代数和物理学之间发展密切而富有成效的相互作用，帮助巩固群的上同性作为一个涉及理论和计算方法的互动主题。

The principal investigator will apply techniques from algebra andtopology to study a number of basic questions in transformation groups, orbifolds, field theory, group cohomology and related topics. Among other things he proposes the following research projects: (1) construction of finite group actions using methods from groupcohomology and homotopy theory; (2) research on representation spaces and K-theory of the braid groups and other discrete groups; (3) algebraictopology of orbifolds: the development and study of topological invariantsassociated to orbifolds; (4) field theory and cohomology of groups: theanalysis of Galois groups using cohomological methods; and (5) computationof the cohomology of low rank simple groups, using a combination oftheoretical and computer-assisted methods.This proposal is interdisciplinary in its conception, as it intendsto combine methods from algebra and topology to address a broad range of problems. Symmetry is a topic of central interest in mathematics,which can be effectively studied using topological methods. Research on mathematical aspects of orbifold string theory is representative of the emerging fusion between geometry and physics. Many important ideas have originated in physics and they consistently lead to the opening of new frontiers in mathematics. On the other hand the cohomology of groups is a subject involving intricate algebraic invariants which very naturally play a role in representation theory, number theory, field theory, etc. Recent techniques use advanced methods and capabilities from computer algebra, and the proposed calculations will serve as benchmarks for further computational as well as theoretical developments in the field--with many potential applications. This proposal will seek to develop close and fruitful interactions between topology, algebra and physics, helping to consolidate the cohomology of groups as an interactive subject involving both theoretical and computational methods.