This award will support mathematical research focusing on three problem areas. The first continues work on quasiconformal maps defined on manifolds. These maps play an essential role in understanding transformations of Euclidean space whose local distortions lie within fixed limits. The extension to manifolds such as the Heisenberg group is new, providing a potentially valuable tool in the harmonic analysis of semi-simple or nilpotent Lie groups. In addition, the manifolds under consideration form boundaries of strongly pseudoconvex domains in several complex variables. It then becomes natural to ask (as was in the real variable case) for conditions on when these quasiconformal maps can be extended. Many of the real-variable methods do not carry over. Nevertheless, some of the main results do. Work will concentrate on isolating conditions which will give the extension result. A second line of research involving symmetric spaces concerns domains formed by symmetric cones and tube domains. At issue is whether or not on can obtain all symmetric tubes by a Jordan algebra construction. If this can be done, then it is likely that current results available for convex cones can be carried over to the non-convex. This will have particular implications for representation theory. The third area of investigation will consider harmonic functions on Riemann surfaces and on discrete structures. Efforts will be made to carry over some of the classical one- variable function theory to complete Riemann spaces. Of special interest is the use of a generalized area integral which can be defined on these spaces. Using this integral work will be done in determining those function spaces for which the area integral gives the same Hardy class as the traditional maximal function.
