Mathematical Sciences: Topics in Dilation Theory

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McCullough will work in the area of dilation theory, a well established part of functional analysis which has made contact with other areas of mathematics such as differential equations and complex function theory. Topics to be investigated include interpolation problems of Caratheodory and Nevanlinna-Pick type, interpolation and commutant lifting theory for general reproducing kernel Hilbert spaces, operator and function theory on multiply-connected domains, as well as the theory of theta functions. 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.