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.
