Group cohomology, rational homotopy theory, and related topics

PI，Dev Sinha，建议研究代数拓扑学中的广泛主题。他将对对称群的上同调做进一步的计算。他将计算等变上同调和分幂结构的K-理论，然后将它们推广到orbillold上同调和K-理论。他将根据上同调、表示理论和群列的适当不变量来开发和计算Hopf环结构。他将Chern-Simons的积分与同伦理论的李余代数模型统一起来，发展出映射的完全有理同伦不变量。他将推广Magnus展开式，并用它来模拟非单连通空间。拓扑学是对形状的基本研究，因此它的根源是几何。作为一门学科，它大致分为考虑基本问题的点集拓扑学，研究特定形状(如我们宇宙的形状)的几何拓扑学，以及最终将形状与数值数据联系起来的代数拓扑学。这不是研究人员在社区之间架起桥梁的典型做法。在他之前资助的研究中，PI应用了从代数拓扑学到纽结理论的方法，这完全是在几何领域。在这项建议中，PI使用了对配置空间(粒子集合)的几何研究的洞察力，以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中的陈-西蒙斯理论联系起来，并发展出一种“字母链接”理论来回答群论中的基本问题。

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.