Group cohomology, rational homotopy theory, and related topics

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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.