Equivariant Algebra, Orientations, and Wilson Spaces

The project sits at the heart of algebraic topology: computing invariants to understand spaces. This is accomplished by systematically building 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. Starting from foundational work of Quillen, algebraic and algebraic geometry data give rise to new invariants for spaces with striking properties. This project combines this classical thread with much more recent developments coming from a theory called equivariant algebraic topology. This theory 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. This extra structure provides more nuanced computations, giving more information about how the classically described invariants change under symmetries. The subject has experienced a renaissance recently dues to the celebrated solution by the PI, Hopkins, and Ravenel to the Kervaire Invariant One problem, one of the oldest outstanding problems in algebraic topology. The solution introduced a host of new constructions and techniques that have striking ramifications in classical and equivariant algebraic topology, and the pieces of this project focus on unpacking some of these new constructions and describing what they mean for algebraic topology in general. Many features of the project also focus on supporting students and early-career researchers and on creating opportunity for collaborations. The PI will develop tools to help mathematicians build and run workshops for graduate students and postdocs, providing entry points into other areas of mathematics similar to that in topology. The PI is also working with their university to build a Geometry Lab and to expand their seminar on art and mathematics into a full course with materials online for self-study. Finally, the PI intends to co-organize a number of international conferences and workshops for researchers in the field. Modern stable homotopy theory heavily utilizes the fact that the stable homotopy category behaves like a derived category of modules. The ground ring here is the sphere spectrum, and computing its homotopy groups is one of the overarching goals in the subject. The problem can be approached by first looking p-locally, and we can pass to the p-local stable homotopy category. Here algebraic geometry provides a further refinement via the theory of formal groups, a cornerstone of algebraic topology. Recent developments by the PI and collaborators have provided a suite of newly accessible computations exploring the connections between equivariant algebra and geometric problems. The PI will build a robust foundation for equivariant algebra in a very generally graded setting, showing how to lift many classical tools and problems to this context. The project also focuses on using the slice spectral sequence to analyze families of orientable representations in classical and equivariant homotopy theory. These themes connect in the core project: a study of an equivalence between orientations by norms of Real bordism and a kind of coordinate on a formal Mackey functor. Finally, the PI will revisit Wilson spaces from an equivariant perspective, showing versions of the main results in Wilson's thesis and building a host of new equivariant commutative rings with nice slices. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.