Motivic homotopy theory

My research revolves around the interaction between homotopy theory and algebra. One of my primary interests is motivic homotopy theory, which is a blend of classical homotopy theory and algebraic geometry. The basic goal of this subject is to solve problems in algebraic geometry using traditional methods of algebraic topology.

Currently, my main interest is a long-term project to make fundamental computations in motivic homotopy theory. This research is supported by a grant from the National Science Foundation.  I work extensively with the motivic version of the Adams spectral sequence, especially over C and R.  Massey products and Toda brackets are the essential computational tools that I use to deduce information about the behavior of this spectral sequence, and ultimately about stable homotopy groups.  Computer calculations play a major role in this project.  My collaborators on this project are Eva Belmont, Dan Dugger, Bogdan Gheorghe, Bert Guillou, Hana Kong, Achim Krause, Guozhen Wang, and Zhouli Xu.