Computations in Stable and Unstable Equivariant Chromatic Homotopy

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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. Starting from foundational work of Quillen, algebraic and algebraic geometry data like formal groups gives rise to new invariants for spaces with striking properties. This project combines this classical thread with much more recent developments coming from equivariant algebraic topology. "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. This extra structure provides more nuanced computations, giving more information about how the classically described invariants change under symmetries. Equivariant algebraic topology has experience a renaissance recently dues to the 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 problems in this project focus on unpacking some of these new constructions and describing what they mean for algebraic topology in general. Many of the projects focus on diversity in STEM. The PI is currently developing tools to help others build a conference series for graduate students and develop their own conferences for younger researchers to attract them to a field. The PI is also in discussions with an HBCU about building more direct connections between their students and the PI's institutions, starting with electronic seminars to introduce students to active researchers in algebraic topology. The goal of these collaborations is to have more students from underrepresented groups enter and succeed in graduate programs in algebraic topology. Finally, the PI has developed and continues to refine a First Year seminar on "Women in Math." The seminar connects students with female mathematicians, allowing the students the opportunity to hear about their research and experience. 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 themes 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. The current approach to understanding monochromatic homotopy is via certain homotopy fixed points computations. Computing the homotopy groups of fixed points and homotopy fixed points is very difficult in general. 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, this is an extremely efficient tool. For larger groups, computations are still tractable but much more mysterious. In all cases, many of the conceptual tools from non-equivariant homotopy are not available. Many of the techniques developed for stable equivariant homotopy can also be applied unstably. This gives a natural and geometric notion of "even" which refines the ordinary one non-equivariantly and which encompasses spaces related to Real bordism and its norms. The PI expects to see a close connection between unstable even spaces, various orientations by norms of Real bordism, and Mackey functor objects in algebraic geometry. 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.