Mathematical Sciences: Realization of Solvable Lie Groups inL 2 Cohomology Spaces and the Structure of Homogeneous Domains in Cn

本课题是李群表示理论的数学研究。后者的一个合适的例子是球体的旋转群。这样的群很重要，因为它们以对称群的形式出现在数学的许多领域(例如，几何、微分方程、代数数论、数学物理)。表象理论允许人们在解决问题时利用对称性。更具体地说，彭尼教授将探索表示理论如何影响复杂分析和几何中的基本问题。其中一个问题涉及平方可积上同调空间中表示的实现。这与n维复空间中无界齐次区域的结构理论有关。其他需要研究的问题包括Koszul区域的Laplace-Beltrami算子的谱与区域几何的关系、这类算子的可解性以及这类区域上调和函数的边界理论。

This project is mathematical research in the representation theory of Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics ( e.g. geometry, differential equations, algebraic number theory, mathematical physics ) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. More specifically, Professor Penney will explore how representation theory impinges on fundamental questions in complex analysis and geometry. One such question concerns the realization of representations in square integrable cohomology spaces. This is related to the structure theory of unbounded homogeneous domains in complex n-space. Other questions to be investigated include the relation of the spectrum of the Laplace - Beltrami operator of a Koszul domain to the geometry of the domain, the solvability properties of such operators, and the boundary theory of harmonic functions on such domains.