Mathematical Sciences: Topics in Dilation Theory

McCullough将在膨胀理论领域工作，膨胀理论是泛函分析中建立良好的一部分，与微分方程和复函数理论等其他数学领域有联系。研究的主题包括Caratheodory和Nevanlinna-Pick型的插值问题，一般再现核Hilbert空间的插值和交换子提升理论，多连通域上的算子和函数理论，以及函数理论。算符理论是研究矩阵的无限维推广的数学部分。特别地，当限制在有限维子空间中时，算子具有通常的线性性质，因此可以用矩阵表示。算子理论的中心问题是对满足附加条件的算子进行分类，这些附加条件是根据相关算子（如伴随算子）或根据基础空间给出的。算子理论是许多数学的基础，也是数学在其他科学中的许多应用的基础。

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.