Mathematical Sciences: Geometric Problems in Several ComplexVariables
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Two general directions of mathematical research in the                     field of several complex varibles characterize this project.                    Both may be viewed as geometric in their statement, but analytic                in the methods employed.                                                            The first question is the result of a long-term effort to                   understand the nature of proper mappings between balls in spaces                of several complex variables.  A holomorphic mapping beween                     discs in one dimension (continuous up to the boundary) can only                 be achieved by a finite Blaschke product.  In higher dimension,                 the only (proper) maps were shown to be automorphisms.  The                     question was reopened when it was discovered that when the range                was a ball of higher dimension, there were distinctly new and                   interesting proper maps from the lower to the higher dimensional                balls.  They may not be smooth up to the boundary, but if they                  are then one can say when they must be rational.  When the map                  is actually polynomial, then it factors completely analogously                  to the finite Blaschke product.  The thrust of this work will be                to take up the classification of those rational proper maps                     which are not polynomial.  It is believed that they too admit a                 factorization into simple component parts.                                           The second part of the work concerns the geometry of real                  hypersurfaces in complex n-dimensional space and Cauchy-Riemann                 manifolds.  This is more technical in its description.  The goal                is to formulate some of the extrinsic notions of multiplicities                 and orders of contact for real hypersurfaces in an intrinsic                    manner that will extend to CR manifolds.  Questions about the                   geometry of real hypersurfaces are part of a larger effort to                   understand surface geometry near points where the Levi form is                  degenerate.  Some of the most promising research taking place in                several complex variables at the present time centers on domains                whose boundaries have degeneracies such as those studied here.
{'FirstName': 'John', 'LastName': "D'Angelo", 'PI_MID_INIT': 'P', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': "John P D'Angelo", 'EmailAddress': 'jpda@math.uiuc.edu', 'PI_PHON': '2173336406', 'NSF_ID': '000221792', 'StartDate': '06/01/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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