Computations in Equivariant Homotopy and Algebraic K-Theory

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AbstractAward: DMS 1207774, Principal Investigator: Michael A. HillThis project seeks to broaden our knowledge of equivariant computations and, by universality of certain maps, of the homotopy groups of spheres. Equivariant homotopy calculations are notoriously difficult, hybridizing representation theory and classical stable homotopy theory in beautiful, and sometimes mysterious, ways. The recent solution by the principal investigator, Hopkins, and Ravenel to the Kervaire Invariant One problem introduced several new tools and techniques to equivariant homotopy, rigidifying earlier homotopical work and generalizing naturally occurring filtrations. In this project, the PI intends to use these new, exciting tools to continue making computational inroads in equivariant homotopy. There are two main approaches: computing equivariant homotopy groups of the families of spectra introduced by the PI, Hopkins, and Ravenel, and reconceptualizing parts of the cyclotomic trace approaches to algebraic K-theory. The former uses slice filtration techniques to tackle spectra like the Hopkins-Miller spectra, allowing a direct attack on the stable homotopy groups of spheres from a non-traditional approach. The latter builds on the norm machinery to rewrite the classical constructions in the equivariant approaches to algebraic K-theory, recasting them in computationally more amenable forms.The project addresses directly the heart of algebraic topology: computing numbers and invariants to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. Equivariant algebraic topology remembers a collection of symmetries inherent in a space as part of the data, systematically grouping spaces with the same symmetries, and the numbers and invariants produced must reflect this. Remembering the extra structure makes richer, but more complicated, computations, and it allows us to tease apart otherwise interconnected problems. For example, using equivariant methods, the principal investigator, Hopkins, and Ravenel solved the Kervaire Invariant One problem, the oldest outstanding problem in algebraic topology with roots dating back to the 1930s. This in turn gave information about how we can build spaces out of simpler ones like spheres. This project aims to build on the techniques developed in the solution, tackling other computational problems in algebra and topology.