Bill Helton's current interest is in equations and inequalities for functions having matrix variables. The functions are typically (noncommutative) polynomials or rational functions. Such functions are central to linear systems problems, which are specified entirely by a signal flow diagram and L^2 performance specs on signals, to free probability (random matrices), and quantum information theory. In short, Bill and his team are developing a noncommutative (free) algebraic and real algebraic geometry. An elegant theory emerged last year: the behavior of singularities (nullstellensatz) fell into place.
Today's theory operates below the generality seen in linear systems theory, so implications there are only by analogy. However, one consequence for random matrices gives formulas for the asymptotic eigenvalue densities of a rational function of random matrices. They are the main providers to Mathematica of noncommuting algebra capability.
Additionally, Bill has worked with Vinnikov on the scope of LMI techniques and with Jiawang Nie on scope of SDP techniques. Bill has also worked on H^\infty control theory and noncommutative geometry.
- Geometry of Polynomials, Spring 2019. Visiting Scientist.