Helton will work to extend the classical theory of interpolation of matrix-valued functions, and the branch of operator theory called the commutant lifting, in several different ways: to extend the basic linear theory, to develop such a theory for nonlinear generalizations of multiplication operators and operator models, and to develop function theoretic extensions and their numerics. Another major effort will be to extend this type of mathematics to nonlinear operators, control systems, and circuits. Operator theory is that part of mathematics that studies the infinite dimensional generalizations of matrices. In particular, when restricted to finite dimensional subspaces, an operator has the usual linear properties, and thus can be represented by a matrix. The central problem in operator theory is to classify operators satisfying additional conditions given in terms of associated operators (e.g. the adjoint) or in terms of the underlying space. Operator theory underlies much of mathematics, and many of the applications of mathematics to other sciences.
