RUI: equivariant and motivic loop spaces

The National Science Foundation Division of Mathematical Sciences awarded The Reed Institute $270,000 on August 15, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to advance homotopy theory through equivariant and motivic methods.

The research develops algebraic and combinatorial tools for understanding symmetry in mathematical structures and imports topological techniques into algebraic geometry. Work encompasses three research thrusts investigating diachromatic blueshift in motivic stable homotopy theory, equivariant aspects of homotopy theory, and structural and computational results. One project component applies topological data analysis to detect symmetry in images, with potential applications in image processing, machine learning, and artificial intelligence.

A central focus is undergraduate research training through the Collaborative Mathematics Research Group (CMRG), a summer program at Reed College since 2018 that engages undergraduate students in original research resulting in peer-reviewed publications. This workforce development component supports retention in the mathematical sciences and prepares students for STEM careers.

Performance occurs in Portland, Oregon, with a period running through July 31, 2028. The assistance type is Project Grant. Reed College, a Portland-based private nonprofit educational institution, serves as the recipient. The project advances fundamental mathematical infrastructure supporting modern science while developing early-career mathematical talent aligned with NSF's broader-impact mission.

Abstract

Homotopy theory is the branch of mathematics that studies the properties of shapes and spaces that are preserved under continuous deformation. Over the past century IT has developed into a powerful and abstract toolkit whose ideas now reach across mathematics and into neighboring sciences. This project advances homotopy theory by developing new algebraic and combinatorial tools for understanding symmetry (through equivariant methods) and for importing topological techniques into algebraic geometry (through motivic methods). Progress on these foundational questions strengthens the mathematical infrastructure on which much of modern science depends, serving the national interest by promoting the progress of science. One component of the project also connects topology to the analysis of data and images: IT uses topological data analysis to detect symmetry in images, an approach with potential applications in image processing, machine learning, artificial intelligence and related areas. A central aim of this award is the training of undergraduate researchers. Through the collaborative mathematics research group (cmrg), a summer program the PIS have led at reed college since 2018, undergraduate students carry out original research that has repeatedly resulted in publications in peer-reviewed research journals. By engaging students in genuine research early in their careers, the project develops mathematical talent, supports retention in the mathematical sciences, and helps prepare a skilled stem workforce, advancing national prosperity and welfare. The PIS will investigate equivariant, motivic, and combinatorial aspects of homotopy theory, pursuing structural, computational, and enumerative results across three research thrusts. The first studies diachromatic blueshift in motivic stable homotopy theory: by computing the C2-tate fixed points of real algebraic johnson-wilson theories via the slice spectral sequence, the PIS aim to construct and exploit exotic w_n-periodicity classes and to give a uniform, base-scheme-independent definition of these classes. The second develops the theory of algebras over the genuine equivariant operads of bonventre-pereira, defining such algebras as coefficient systems with additional structure, constructing the associated composition product and model structures, and establishing the expected quillen adjunctions. The third constructs categorical models for equivariant little disks operads, building operads in g-categories whose classifying spaces model the little v-disks operads for finite-dimensional representations v, beginning with the group of order two and extending toward more general groups via iterated monoidal structures and dunn additivity. The methods draw on the slice and homotopy fixed point spectral sequences, motivic and equivariant stable homotopy theory, operad theory, and category theory. Together the projects promise refined structural and computational control over motivic and equivariant phenomena, with applications to exotic periodicity, equivariant spectra with prescribed fixed points, and the construction of loop spaces from categorical data. The research program is integrated with the cmrg, through which undergraduate teams will pursue two accessible projects synergistic with the PIS' work: detecting wallpaper symmetry in images using persistent homology (topological data analysis), and studying the combinatorics of model structures on posets, including enumerative questions and generalizations to infinite lattices. These projects introduce students to contemporary research, strengthen their technical communication, and prepare them for graduate study and research careers. 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.- subawards are not planned for this award.