My research revolves around noncommutative generalizations of real algebra and real algebraic geometry, including:

- ordered algebras and algebras with involution,
- representation theory,
- obstructions to orderability,
- semidefinite programming,
- quadratic and hermitian forms, forms of higher level,
- noncommutative valuation theory.

- Finsler's Lemma for matrix polynomials, Arxiv , submitted
- (with J. William Helton, Scott McCullough, Christopher Nelson) Real Nullstellensatze and *-ideals in *-algebras, Arxiv , To appear in Electronic Journal of Linear Algebra.
- (with Y. Savchuk) Induced quadratic modules in $*$-algebras, Arxiv , To appear in Communications in Algebra.
- On generalized real semiprime ideals, pdf, submitted.

- European Journal of Pure and Applied Mathematics, associate editor
- Journal of Algebraic Statistics, editor
- Journal of Algebra and Computational Applications, editor