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

Penney教授将研究复n空间中在实李群下齐次的一类域。这类是Koszul定义域族，它是通过要求某个双线性形式是非退化的来定义的。这种形式给出了伪kahlerian流形的定义域结构。研究的一个问题是拉普拉斯-贝尔特拉米算子的特征值问题。这已经在一级案例中完成了，Penney教授将把它扩展到更高级别的案例中。这将需要在更高级别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合作，研究可解李群的重陪集空间的几何描述。群论基本上就是对称理论。举一个简单的例子，当所讨论的系统在空间原点位置的变化下是不变的，平移组自然产生。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”。Penney教授将研究在复杂n维空间中作用于特定域的群的表示。

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.