Equivariant Derived Algebraic Geometry

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The project addresses directly the heart of algebraic topology: computing invariants like numbers, groups, and rings to understand spaces. The goal of algebraic topology is to systematically build a connection between algebraic objects like numbers and geometric objects like spaces. This connection allows a two-way flow of information, with algebraic invariants distinguishing spaces and topological methods informing algebraic problems. A beautiful example of the latter is the Goerss-Hopkins-Miller theory of topological modular forms, a way to encode elliptic curves (a fundamental object in algebraic geometry) in topological language. This builds new kinds of elliptic curves and highlights commonalities not visible through ordinary algebra. "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 one to tease apart otherwise interconnected problems. For example, using equivariant methods, the PI, 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 one 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. In particular, the project seeks to explore the interaction between the visible equivariance in settings like topological modular forms arising from underlying algebraic data and the constraints placed by the topology.Modern stable homotopy theory heavily utilizes the fact that the stable homotopy category behaves like a derived category of modules. Here the ground ring is not an ordinary ring but rather a ring spectrum, the sphere spectrum. Work over the last twenty years has described how to do algebraic geometry directly with commutative ring spectra: the theory of derived algebraic geometry. Many of the naturally occurring examples arise as commutative ring spectra with an action of a finite group, so one asks when there is an underlying equivariant commutative ring spectrum which is computationally accessible. This is the main focus of this project, a new area of research called "equivariant derived algebraic geometry". From a computational perspective, the goal is to understand the interplay between the homotopy groups of fixed points of a group action on a spectrum and the underlying homotopy groups of the spectrum. In general, this is a very difficult problem. One of the most exciting new tools developed to solve the Kervaire problem is a general slice filtration, a method which directly computes homotopy groups of fixed points. For Real Landweber exact theories, theories well-rooted in the underlying algebraic geometry, this is an extremely efficient tool. For larger groups, computations are tractable but much more mysterious. One of the goals of the project is to determine when the kinds of spectra arising from equivariant derived algebraic geometry have slices as nice as those for Real Landweber exact theories. Equivariant homotopy is also central to the homotopical approach to algebraic K-theory. Algebraic K groups are also exceedingly difficult to compute, and even knowing whether or not they are zero would settle long-standing number theory conjectures. The primary approach in homotopy is via a tower of spectra, the TR tower, built inductively out of fixed point spectra for topological Hochschild homology. The new equivariant machinery provides alternate, simpler construction of topological Hochschild homology, allowing us to evaluate it on Thom spectra and to build relative versions.