Mathematical Sciences: Topics in Dilation Theory

McCullough将在膨胀理论领域工作， 建立了一部分功能分析， 数学的其他领域，比如微分方程 复变函数理论 调查的主题包括 Caratheodory和Nevanlinna-Pick型插值问题， 一般插值与交换提升理论 再生核Hilbert空间，算子与函数论 关于多连通域以及θ理论 功能协调发展的 算子理论是数学中研究 矩阵的无穷维推广 特别是， 当限制到有限维子空间时，算子具有 通常的线性性质，因此可以表示为 矩阵 算子理论的中心问题是分类 满足附加条件的算子 关联算子（例如伴随算子）或 底层空间 算子理论是数学的基础， 以及数学在其他科学中的应用。

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.