Mathematical Sciences: Commuting Extensions of Certain Differential Operators

Pedersen教授将研究作用于紧支持光滑函数上的一阶偏微分算子的可交换自伴随扩展。这些扩展可能存在，也可能不存在，这取决于给定的域。这个项目的动机部分来自于理解高阶常系数偏微分算子的谱性质的愿望，也来自于交换自伴随扩展的谱与区域几何之间的联系。Pedersen教授还将研究这些思想对与定义域相关的各种拉普拉斯算子的影响。这个项目涉及到微分理论（多变量）和最简单形式的几何之间的联系。Pedersen教授将研究的一个想法是这些微分算子的特征值是如何与n空间中定义它们的区域的几何结构相关联的。

Professor Pedersen will study commuting selfadjoint extensions of first order partial differential operators acting on smooth functions of compact support on some domain. Such extensions may or may not exist depending on the given domain. This project is motivated in part by the desire to understand spectral properties of higher order constant coefficient partial differential operators, and also by the connection between the spectrum of the commuting selfadjoint extension and the geometry of the domain. Professor Pedersen will also investigate the implications of these ideas for the various Laplace operators associated to the domain. This project involves the connection between the theory of differentiation (in many variables) and geometry in its simplest form. One of the ideas which Professor Pedersen will investigate is how the eigenvalues of these differentiation operators are related to the geometry of the domain in n-space on which they are defined.