Mathematical Sciences: Commuting Extensions of Certain Differential Operators

Pedersen教授将研究作用于某些区域上紧支撑的光滑函数上的一阶偏微分算子的交换自伴扩张。这种扩展可能存在，也可能不存在，具体取决于给定域。这个项目的部分动机是希望了解高阶常系数偏微分算子的谱性质，也是因为交换自伴扩张的谱与区域的几何之间的联系。彼得森教授还将研究这些想法对与该领域相关的各种拉普拉斯运算符的影响。这个项目涉及到微分学理论(在许多变量中)和最简单形式的几何学之间的联系。Pedersen教授将研究的想法之一是这些微分算子的特征值如何与定义它们的n空间中域的几何关系。

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.