Group cohomology, rational homotopy theory, and related topics

群上同调、有理同伦理论及相关主题

• 批准号: 1006819
• 负责人: Dev Sinha
• 金额: $ 12.89万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006819&HistoricalAwards=false
• 关键词: Group; cohomology; rational; homotopy; theory

The PI, Dev Sinha, proposes to investigate a wide range of topics in algebraic topology. He will make further calculations in the cohomology of symmetric groups. He will compute equivariant cohomology and K-theory of divided powers constructions and then extend them to orbifold cohomology and K-theory. He will develop and compute Hopf ring structures on cohomology, representation theory and suitable invariants of series of groups. He will unify integrals from Chern-Simons with the Lie coalgebraic model of homotopy theory to develop complete rational homotopy invariants of maps. He will generalize the Magnus expansion and use that to model non-simply connected spaces.Topology is a fundamental study of shape, and thus has its roots in geometry. As a subject it has roughly split into point-set topology which considers foundational questions, geometric topology which studies particular shapes such as that of our universe, and algebraic topology which ultimately relates shape to numerical data. It is not typical for researchers to bridge thees communities. In his previously funded research, the PI applied methods from algebraic topology to knot theory, which is squarely in the geometric realm. In this proposal, the PI is using insight from the geometric study of configuration spaces (collections of particles) to better understand algebraic structures such as symmetric groups and symmetric functions. He also plans to connect algebraic topology with Chern-Simons theory from mathematical physics, and to develop a theory of "linking of letters" to answer basic questions in group theory.

PI，Dev Sinha，提议研究代数拓扑中的广泛主题。他将进一步计算对称群的上同调。 他将计算等变上同调和K-理论的划分权力建设，然后将其扩展到orbifold上同调和K-理论。 他将开发和计算的上同调，代表性理论和适当的不变量的一系列群体的霍普夫环结构。 他将把陈-西蒙斯的积分与同伦理论的李余代数模型统一起来，以开发地图的完整有理同伦不变量。 他将推广马格努斯展开式，并使用它来模拟非单连通空间。拓扑学是形状的基础研究，因此它的根源在几何学。 作为一个主题，它已大致分为点集拓扑考虑的基础问题，几何拓扑研究特定的形状，如我们的宇宙，和代数拓扑最终涉及形状的数值数据。 研究人员在社区之间架起桥梁并不常见。 在他以前资助的研究中，PI应用了从代数拓扑到结理论的方法，这完全属于几何领域。 在该提案中，PI利用配置空间（粒子集合）几何研究的洞察力来更好地理解对称群和对称函数等代数结构。 他还计划将代数拓扑学与数学物理学中的陈-西蒙斯理论联系起来，并发展一种“字母连接”理论来回答群论中的基本问题。