RUI: Motivic, Operadic, and Combinatorial Homotopy Theory

Reed College, Portland OR

Investigators

Abstract

The scope of contemporary homotopy theory — the mathematics of shape and deformation — has expanded dramatically in recent years, and new tools and perspectives are needed to understand its features and potential. The PIs will explore homotopy theory through the lenses of combinatorics (enumerating structures), operads (parametrizing operations), and algebraic geometry (shapes described by polynomial equations). The PIs will also support the participation of a group of undergraduate students in their research by organizing the Collaborative Mathematics Research Group (CMRG). CMRG participants will produce novel research results and also engage their local community in mathematics outreach efforts. The PIs will investigate operadic, combinatorial, and motivic aspects of homotopy theory with the aim of producing structural, enumerative, and computational results. Their projects include the following: (1) Further explore the combinatorics of model structures on finite categories, especially with an eye towards Catalan combinatorics of arbitrary type (in the sense of cluster algebras). (2) Uncover additional structure in the Balmer spectrum of the stable motivic homotopy category by leveraging a recollement with its étale variant. (3) Extend our computational understanding of stable motivic homotopy theory by computing the slice spectral sequence for the Bachmann-Hopkins connective Hermitian image-of-J spectrum. (4) Construct a rigid model for E_n-spaces as commutative monoids in a certain diagram category. 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.