8900234 Rubin Transformations from the space of several complex variables to itself which are holomorphic (in each coordinate) are generally called automorphisms. The automorphisms form a group - the group of all changes of coordinates which preserve analyticity. While these mappings are fully understood in one complex dimension, the size of the group increases dramatically when the dimension increases above one. The main thrust of this continuing work is to understand the structure of the automorphism group. Several new geometric features of the group have been obtained over the past two years, using elementary methods to construct automorphisms which satisfy a variety of prescribed conditions. There remain many specific open questions which now appear to be tractable. Among these is the problem of classifying discrete subsets whose image is an arithmetic progression under some automorphism, (tame sets). It is not known exactly what makes a set tame. Evidence suggests that this property is related to how well the set is uniformly separated. A second conjecture contends that tame sets must lie in the image of a proper holomorphic embedding of one dimensional complex space. Closely related to holomophic embeddings are the holomorphic maps of n-dimensional complex space into itself and the regions which can result as the range of such mappings. These regions are probably best characterized by describing properties of omitted sets. For instance, work will be done in an effort to determine whether such regions in two dimensions can omit two nonparallel lines. Work will also be done investigating the smoothness of the boundary of the range. Other work will consider inner functions defined in the two-dimensional ball (shown to exist during the early 1980's). Specifically, one would like to know if pairs of independent inner functions can be determined. These would provide measure-preserving maps from the boundary of the ball to the 2-torus and yield isometries from the Hardy space of the bidisc to the corresponding space in the ball.
