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.  It is a surprising geometric fact that spheres of different dimensions can fit together in only a few different ways.  Enumerating these combinations of spheres is one of the fundamental questions of stable homotopy theory.  This chart represents one possible approach to the problem.  Each dot represents a particular combination of spheres.  The lines indicate certain relationships between these different combinations.  You can view a presentation that I gave on this topic in January 2014 at the Mathematical Sciences Research Institute in Berkeley, California, USA. 
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 Dan Dugger, Bert Guillou, and Zhouli Xu.
In conjunction with Dan Dugger, I have generalized results about sums-of-squares formulas over fields of characteristic zero so that they apply over arbitrary fields of characteristic not equal to 2. The basic idea is to replace techniques involving cohomology theories for topological spaces with techniques involving algebraic cohomology theories.
I have also spent some time with Cayley-Dickson algebras.  These are finite-dimensional non-associative algebras that generalize the real numbers, the complex numbers, the quaternions, and the octonions.  In conjunction with Dan Dugger, Daniel Biss, and Dan Christensen, we have attempted to classify the zero-divisors in Cayley-Dickson algebras.
I am also interested in abstract homotopy theory (i.e., model categories), especially with respect to homotopy theories for pro-objects.
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