Equivariant and Chromatic Stable Homotopy Theory

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This research project explores questions in algebraic topology, a branch of mathematics that concerns shapes in higher dimensions. Despite its abstract nature, algebraic topology has proven to be useful in a variety of applications, including theoretical physics, where the subject arises naturally in attempts to reconcile general relativity with quantum mechanics, and data science, where the subject has had considerable impact in the analysis of large data sets. A fifty-year-old question in the field known as the Kervaire invariant problem was solved in 2009; the answer to the question at the heart of the problem was the opposite of what most experts had expected, and surprising new techniques, potentially useful in other areas, were required in the proof. The research project aims to amplify this discovery and adapt it to further applications.The principal investigator plans to follow up on this advance in two ways. First, a book in progress is designed to make the solution methods accessible to graduate students and other interested non-experts in the field, amplifying the mathematical infrastructure in equivariant homotopy theory and category theory for the Kervaire invariant problem with illustrative examples and explanations. Second, the tools developed to solve the Kervaire invariant problem are being adapted to further applications. In particular, there is a counterpart to the problem for each prime number. The recent solution of the original, geometrically motivated, problem was for the prime 2. The algebraic analog for primes 5 and larger had been solved in the late 1970s. The algebraic problem remains open for the prime 3, and the principal investigator has a plan for solving it. In addition, the surprising nature of the solution to the 2-primary problem raises more questions than it answers, implying in particular that certain predicted patterns in the homotopy groups of spheres cannot occur. The question of what might replace them is wide open.