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

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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.