Mathematical Sciences: "The Structure and Function Theory of Homogeneous Domains in Cn"

Penney教授将研究复n空间中的一类区域，这些区域在实李群下是齐次的。这个类是Koszul域的族，它是通过要求某个双线性形式是非退化的来定义的。这种形式给定域提供了伪卡勒流形的结构。一个需要研究的问题是Laplace-Beltrami算子的特征值问题。这已经在第一级案例中完成了，彭尼教授将把这一点扩展到更高的级别。这将需要在更高等级的情况下对这些域的结构进行更多的工作。Penney教授还将继续与B.Currey在可解李群的双陪集空间的几何描述方面的合作。群论基本上是对称论。举一个简单的例子，当所讨论的系统在空间原点的位置改变时是不变的，自然就产生了翻译群。虽然群是抽象的对象，但特殊情况需要对称群的具体实现或“表示”。彭尼教授将研究复杂n维空间中作用于某些区域的群的表示。

Professor Penney will study a certain class of domains in complex n-space which are homogeneous under a real Lie group. This class is the family of Koszul domains which are defined by requiring that a certain bilinear form be non-degenerate. This form gives the domain the structure of a pseudo-Kahlerian manifold. One problem for study is the eigenvalue problem for the Laplace- Beltrami operator. This has been done in the rank one case and Professor Penney will extend this to higher rank. This will necessitate more work on the structure of such domains in the higher rank case. Professor Penney will also continue his collaboration with B. Currey on the geometric description of double coset spaces for solvable Lie groups. Group theory is basically the theory of symmetry. To take a simple example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group. Professor Penney will be studying representations of groups acting on certain domains in complex n-dimensional space.