NSF Award
2427220
Algebraic topology is a field of mathematics that involves using algebra and category theory to study properties of geometric objects that do not change when those objects are deformed. A central challenge is to classify all maps from spheres to other spheres, where two maps are considered equivalent if one can be deformed to the other. The equivalence classes of these maps are called the homotopy groups of spheres, and collectively they form one of the deepest and most central objects in the field. Historically, much important theory has arisen out of attempts to compute more homotopy groups of spheres and understand patterns within them. This project involves furthering knowledge of the homotopy groups of spheres, using old and new techniques as well as computer calculations. The project also involves studying an analogue of these groups in algebraic geometry; this falls under a relatively new and actively developed area called motivic homotopy theory, which applies techniques in algebraic topology to study algebraic geometry. The broader impacts of this project center around supporting the local mathematics community through mentoring and promoting diversity. The principal investigator will help build the nascent homotopy theory community at the university and promote women and minorities in the subject through seminar organization and mentoring.<br/> <br/>One of the main planned projects is a large-scale effort to compute the homotopy groups of spheres at the prime 3 in a range, using old and new techniques such as the Adams-Novikov spectral sequence as well as infinite descent machinery. This work will be aided by computer calculations, which short-circuits some of the technical difficulties encountered in previous attempts. Another main group of projects concerns computing the analogue of the stable homotopy groups of spheres in the world of R-motivic homotopy theory. This represents a continuation of prior work of the PI and collaborator; the plan is to supplement the techniques used in that work with computer calculations and a new tool, the slice spectral sequence. A third project concerns theory and spectral sequence computations aimed at computing the cohomology of profinite groups such as special linear groups and Morava stabilizer groups.<br/><br/>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.
