Mathematical Sciences: Several Complex Variables
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work will be done on geometric questions concerned with domains                 and functions of several complex variables.  The principal                      investigator's work over the course of several grant periods                    has been concerned with a type of convexity related to the                      study of function algebras.  The class of polynomials defined                   on a subset of n-dimensional complex space extend to a larger                   subset, the polynomial convex hull, in such a way that they do                  not increase in size.   When the hull possesses analytic                        structure, related function algebras have a concrete                            realization.  For example, if the originating set is connected,                 the hull is analytic (or void) but when the connectedness                       hypothesis is dropped, the conclusion may fail.  But all                        examples of this type have infinite area.  Work will be done in                 determining the relationship between area and the nature of the                 hull.  There are at present two approaches which will be tried.                                                                                                 Another line of investigation will pursue extensions of a                       remarkable result proved recently by the principal                              investigator:  A holomorphic function of one complex variable,                  defined in a disc, whose global cluster set has finite linear                   measure, extends continuously to the boundary of the disc.                      What remains to be resolved is that of finding the optimal                      result.  That, in itself, requires formulation.                                                                                                                 Recent work on hulls in the space of two complex variables in                   collaboration with John Wermer of Brown University will                         continue.  They have had particular success                                     analyzing hulls of sets which lie over one-dimensional discs.                   That is, the first coordinate belongs to some fixed disc.  Sets                 with this structure fiber over the disc.  When the fibers                       themselves have some geometric character, the hull can be                       described.  For example, when the fibers are convex, the hull                   is built from graphs of bounded analytic functions.  This                       result led to a new proof of the one-dimensional corona                         theorem.  Thus, it is natural to consider how much of what has                  been done can be carried over to n-dimensional balls.
{'FirstName': 'Herbert', 'LastName': 'Alexander', 'PI_MID_INIT': 'J', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Herbert J Alexander', 'EmailAddress': 'U22330@UICVM', 'NSF_ID': '000279225', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Illinois at Chicago', 'CityName': 'CHICAGO', 'ZipCode': '606124305', 'PhoneNumber': '3129962862', 'StreetAddress': '809 S MARSHFIELD AVE M/C 551', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Illinois', 'StateCode': 'IL', 'CONGRESSDISTRICT': '07', 'CONGRESS_DISTRICT_ORG': 'IL07', 'ORG_UEI_NUM': 'W8XEAJDKMXH3', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF ILLINOIS', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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