Evenness in Modern and Classical Homotopy Theory

In this project, the PI investigates the field of mathematics known as algebraic topology. Topology is the study of shapes. Arising out of the age-old study of geometry, topology lies at the essence of practical problems - from analyzing DNA folding to planning robot motion in factories to understanding the intricacies of gravity and space. Through algebraic topology, the PI attacks these problems using powerful algebraic tools; in particular, the PI focuses on the ubiquitous phenomenon of evenness, which has arisen as a uniting force between algebraic topology and the related fields of arithmetic geometry and representation theory. In addition, the project will have broader impacts in mentorship and professional training for graduate students and postdocs at Northwestern, the PI’s home institution, funding various activities to jump-start their careers. At a more specific level, the project aims to study two particular directions at the intersection of algebraic topology and arithmetic geometry: the prismatization of commutative ring spectra via topological Hochschild homology, and classical questions in unstable homotopy theory and Wilson spaces. These are united by the theme of evenness, and in particular, the way that commutative ring spectra with even homotopy groups play a key role in both of these questions. To approach these questions, the PI will use recently developed techniques from higher algebra, higher category theory, chromatic homotopy theory, and algebraic K-theory. As such, the project has the potential not only to significantly advance the state-of-the-art in the study of prismatic cohomology, but open whole new avenues of research in unstable homotopy theory. 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.