Mathematical Sciences: Holomorphic Mappings in Several      Complex Variables
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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.
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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