Equivariant Methods in Chromatic Homotopy Theory

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The spheres are among the simplest geometric objects, and they are the building blocks of more complicated topological spaces. The homotopy groups of spheres are collections of continuous functions between spheres considered up to certain deformations. These groups hold fundamental information about maps between topological spaces and have deep connections to number theory, differential topology, and geometric topology. However, despite their simple definition, the homotopy groups of spheres are extremely difficult to compute. To better understand these groups, chromatic homotopy theory is a powerful tool that organizes theory and computations by analyzing the algebraic geometry of smooth one-parameter formal groups. The moduli stack of formal groups has a stratification by height, which corresponds in the stable homotopy category to localizations with respect to the Lubin-Tate theories. The Lubin-Tate theories give rise to higher periodicity in the stable homotopy groups of spheres, and studying them is one of the most important areas of research in chromatic homotopy theory. Starting from Hill-Hopkins-Ravenel's resolution of the Kervaire invariant problem, the newly developed equivariant machinery offers new methods to attack problems in chromatic homotopy theory that were notoriously difficult to approach via classical methods. The planned research explores the connections between equivariant homotopy theory and chromatic homotopy theory, and uses cutting-edge equivariant technology to produce state-of-the-art computations in chromatic homotopy theory.The principal investigator will study periodicity phenomena in the stable homotopy groups of spheres through the lens of equivariant and chromatic homotopy theory. In current and ongoing projects, the PI establishes the first known connection between the obstruction-theoretic actions in chromatic homotopy theory and the geometry of complex conjugations. Using this newly discovered connection, the PI will study Lubin-Tate theories as equivariant spectra and use equivariant machinery developed by Hill-Hopkins-Ravenel to produce higher chromatic height computations at the prime 2. This project aims to deepen the connection between equivariant and chromatic homotopy theory by proving Periodicity, Gap, and Detection theorems for norms of Real bordism theories and fixed points of Lubin-Tate theories. The PI will carry out more chromatic computations to study the last open case of the Kervaire invariant problem. The PI will also investigate Hurewicz images, prove general differential patterns, and exhibit transchromatic phenomena in the slice spectral sequences of Real bordism theories and Lubin-Tate theories across different groups and heights.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.