Group cohomology, rational homotopy theory, and related topics

PI, Dev Sinha，提议调查代数拓扑中的广泛主题。他将对对称群的上同调作进一步的计算。他将计算分幂结构的等变上同调和k理论，然后将其推广到轨道上同调和k理论。他将在上同调、表示理论和群系列的合适不变量的基础上发展和计算Hopf环结构。他将把chen - simons的积分与同伦理论的Lie共代数模型统一起来，发展映射的完全有理同伦不变量。他将推广马格努斯展开，并用它来模拟非单连通空间。拓扑学是一门关于形状的基础研究，因此它植根于几何学。作为一门学科，它大致分为考虑基础问题的点集拓扑，研究特定形状（如我们的宇宙）的几何拓扑，以及最终将形状与数值数据联系起来的代数拓扑。研究人员在这些群体之间架起桥梁并不典型。在他之前资助的研究中，PI将代数拓扑的方法应用到结理论中，这完全属于几何领域。在这个提议中，PI利用构型空间（粒子集合）的几何研究来更好地理解代数结构，如对称群和对称函数。他还计划将代数拓扑与数学物理中的chen - simons理论联系起来，并开发一种“字母连接”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.