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

Penney教授将研究某一类域， 复n-空间，它们在真实的李群下是齐次的。 这 类是Koszul域的族，其定义如下： 要求某个双线性形式是非退化的。 这 形式给域的结构的一个伪Kahlerian流形。 研究的一个问题是拉普拉斯的本征值问题， 贝尔特拉米接线员 这已经在秩1的情况下完成， Penney教授会把这个扩展到更高的级别。 这将 有必要对这些领域的结构进行更多的工作， 更高级的案子 彭尼教授也将继续他的 与B合作。关于二重的几何描述 可解李群的陪集空间 群论基本上是对称性理论。 采取 一个简单的例子，当所讨论的系统在一个 空间原点位置的变化， 翻译自然会出现。 虽然组是抽象对象， 具体情况需要具体实现， 对称群的“表示”。彭尼教授将 研究作用于某些域的群的表示， n维复杂空间

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.