Cohomology and Actions of Finite Groups

有限群的上同调和作用

• 批准号: 0243588
• 负责人: Alejandro Adem
• 金额: $ 13.54万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0243588&HistoricalAwards=false
• 关键词: Cohomology; Actions; Finite; Groups

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.

主要研究者将应用代数和拓扑学的技术来研究变换群、orbifolds、场论、群上同调和相关主题中的一些基本问题。 除其他事项外，他提出了以下研究项目：（1）建设有限群行动使用的方法，从groupcohomology和同伦理论;（2）研究的代表性空间和K-理论的辫子群和其他离散群体;（3）algebraictopology的orbifolds：发展和研究的拓扑invariantsassociated orbifolds;（4）领域理论和上同调的群体：本文的主要内容包括：（1）用上同调方法分析伽罗瓦群;（2）用理论方法和计算机辅助方法相结合计算低秩单群的上同调。对称性是数学中的一个中心问题，可以用拓扑方法进行有效的研究。对轨道弦理论的数学方面的研究是几何学和物理学之间新兴融合的代表。许多重要的思想都起源于物理学，它们不断地导致数学新领域的开辟。另一方面，上同调的群体是一个涉及复杂的代数不变量的主题，很自然地发挥作用，在表示论，数论，场论等最近的技术使用先进的方法和能力，从计算机代数，和建议的计算将作为基准，进一步计算以及理论发展领域-有许多潜在的应用。这个建议将寻求发展密切和富有成效的拓扑学，代数和物理学之间的相互作用，帮助巩固群的上同调作为一个互动的主题，涉及理论和计算方法。