Cohomology and Actions of Finite Groups

主要的研究人员将应用代数和拓扑学的技巧来研究变换群、或支流、场论、群上同调及相关主题中的一些基本问题。在其他方面，他提出了以下研究项目：(1)使用群上同调和同伦理论的方法构造有限群作用；(2)辫子群和其他离散群的表示空间和K-理论的研究；(3)orbilolds的代数拓扑学：与orbiloles有关的拓扑不变量的发展和研究；(4)群的场论和上同调：使用上同调方法分析Galois群；以及(5)用理论和计算机辅助相结合的方法计算低阶单群的上同调。这一建议的概念是跨学科的，因为它试图结合代数和拓扑学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，可以用拓扑学的方法有效地研究它。对奥布里奥德弦理论的数学方面的研究是几何学和物理学之间正在出现的融合的代表。许多重要的思想起源于物理学，它们不断地导致数学领域的新领域的开辟。另一方面，群的上同调是一个涉及复杂的代数不变量的主题，这些不变量非常自然地在表示论、数论、场论等中发挥作用。最近的技术使用了计算机代数的先进方法和能力，所提出的计算将作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。这项提议将寻求在拓扑学、代数和物理学之间发展密切和富有成效的相互作用，有助于巩固群的上同调作为一种涉及理论和计算方法的交互学科。

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.