RUI: Higher Structures in Stable, Equivariant, and Motivic Homotopy Theory

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This research investigates the interface between three branches of contemporary mathematics: higher categories, homotopy theory, and algebraic geometry. Loosely speaking, these subjects study structure, deformations, and polynomials, respectively. The projects undertaken via this award link these concepts in novel ways, seeking to expose new theorems. The Principal Investigors will also support the participation of a diverse group of undergraduate students in their research by organizing summer research groups and leading senior theses related to these projects.The PIs' approach to their research program is five-fold. First, they will jointly study the fashion in which spectral Mackey functors of an absolute Galois group map to "spectral Gysin functors" (constructed from Garkusha-Panin-Voevodksy's theory of framed correspondences) which model motivic spectra. Second, they will study the eta-periodic motivic sphere spectrum via connective Witt K-theory. Third, they will explicate further structure in the Balmer spectrum of the stable motivic homotopy category via comodules over the Lazard Hopf algebroid. Fourth, they will construct equivariant infinite loop space machines which handle pairings of genuine G-permutative categories. And finally, they will study the homotopy theory of Picard 2-groupoids and stable 2-types, and the interaction between the two.