Mathematical Sciences: Linear Fractional Models and Composition Operators

Bourdon教授将研究复平面的单位圆盘到自身的全纯映射的迭代。该项目涉及找到有问题的地图的线性分数模型。研究将在解析函数的希尔伯特空间上的复合算子的框架内进行。这项工作的一个主要组成部分将涉及使用计算机来产生猜想，然后用分析方法证明它们。项目的一部分将在本科生的积极参与下完成。希尔伯特空间算子本质上是复数的无限矩阵。这些运算符在应用科学的各个领域都有应用，在纯数学中也是如此。波登的研究是试图理解这类物体的某些重要类别。

Professor Bourdon will investigate the iteration of holomorphic maps of the unit disk of the complex plane into itself. The project concerns finding a linear fractional model of the map in question. The investigation will take place in the framework of composition operators on a Hilbert space of analytic functions. A major component of the work will involve the use of a computer to generate conjectures and then prove them analytically. Part of the project will be done with the active participation of undergraduates. Hilbert space operators are essentially infinite matrices of complex numbers. These operators have applications in every area of applied science as well as in pure mathematics. Bourdon's research is an attempt to understand certain important classes of such objects.