These projects aim to address major problems in the field of algebraic topology. Topology is the study of geometry where you identify one geometric object with another if one can be deformed into the other. The goal of algebraic topology is to ascribe discrete algebraic invariants to these geometric objects to distinguish their topological types. In this way, distinguishing geometric objects is reduced to algebraic computations. Understanding the topological type of geometric objects is a fundamental act of scientific/mathematical inquiry, comparable to the study of prime numbers, or the classification of the fundamental particles that constitute matter and carry forces. Topological computations have recently been applied to solve problems in solid state physics. Also, data involving the interrelation of a large number of variables naturally traces out a geometric object in a high dimensional space. The study of such data-sets using algebraic topology is the subject of the new and active field of topological data analysis. The focus of these projects is in the interaction of classical algebraic topology, equivariant homotopy theory, and motivic homotopy theory. Equivariant homotopy theory is the study of the topology of symmetry, whereas motivic homotopy theory is the study of the topology of solutions to systems of polynomial equations. Recent years have witnessed a dazzling array of progress in the field of algebraic topology through the importation of equivariant and motivic methods. Broader impacts of these projects include work with graduate students, a summer math research program, a directed reading program, and a bridge program for entering graduate students.
Particular projects involve investigating novel structures in equivariant and motivic stable homotopy theory, and seek to leverage these structures to give new approaches to some long outstanding problems in classical stable homotopy theory. The telescope conjecture will be investigated using a tower of spectra which appeared in the Hill-Hopkins-Ravenel solution of the Kervaire Invariant One Problem. Recent work of Pstragowski and Gheorghe-Isaksen-Krause-Ricka gives a synthetic construction of complex motivic stable homotopy theory. The principal investigator plans to extend this to the real motivic context using equivariant homotopy theory. Another project uses equivariant chromatic homotopy theory to study the interaction of classical chromatic homotopy theory with the Tate construction, with the aim of making progress on the chromatic splitting conjecture. Recent work of Barthel-Schlank-Stapleton gives a means of studying stable homotopy theory at generic primes using ultra-filters. Computations in this context will be investigated using Drinfeld Modules.
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.
