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 used to be interested in abstract homotopy theory (i.e., model categories), especially with respect to homotopy theories for pro-objects. However, I am no longer actively working on problems in this field.