Mathematical Sciences: Planar Dynamical Systems

首席研究员将继续他对动力系统的研究。特别是，他将研究整函数的动力学，与牛顿方法有关的映射，以及一些反全纯映射。这些映射的动力学将被应用于试图理解移位映射的自同构的问题。这些技术还将应用于实动力学中感兴趣的其他族，包括Henon映射族、标准族和在散射理论中出现的映射族。动力系统在数学的各个子领域和数学建模中扮演着越来越重要的角色。本项目要研究的应用之一是在散射理论中的应用。散射论是数学的一个分支，它试图通过描述算符对函数的作用来重构算符。它是物理学中的一个基本工具。不同方法的融合在解决许多数学和科学问题方面发挥着重要作用；这个项目是工作中跨学科研究的一个例子。

The principal investigator will continue his research on dynamical systems. In particular he will study the dynamics of entire functions, maps associated with Newton's method, and some antiholomorphic maps. The dynamics of these maps will be applied to the problem of trying to understand the automorphisms of the shift map. These techniques will also be applied to other families of interest in real dynamics including the Henon map, the standard family, and a family of maps which arise in scattering theory. Dynamical systems are playing an increasing role in various subfields of mathematics and in mathematical modeling. One of the applications to be studied in this project is an application to scattering theory. Scattering theory is the branch of mathematics which attempts to reconstruct an operator from a description of the action of the operator on functions. It is a basic tool in physics. The amalgamation of different methods is playing an important role in the solution of many mathematical and scientific problems; this project is one example of cross-disciplinary research at work.