Mathematical Sciences: Harmonic Measure on Riemann Surfaces
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff'}
Work on this project relates investigations into properties                of analytic functions with the geometry of Riemann surfaces.                    While this research continues much of the same theme represented                in earlier work, the introduction of the tools of Brownian motion               add a new dimension to it.  The underlying approach to problems                 addressed here is one of constructing Riemann surfaces which must               be image surfaces of analytic functions to ensure that the                      mappings have some prescribed properties, especially of a                       geometric nature.                                                                    An example of the type of application possible is that of                  determining and locating harmonic measure by using exit times of                Brownian motion.  If harmonic measure exists, it is supported on                accessible boundary points.  On the other hand, one constructs                  analytic functions by building Riemann surfaces which project                   onto the plane and then combine with a Riemann mapping function.                The difficult part of three constructions is to verify that the                 resulting functions have certain properties.  Here Brownian                     techniques are expected to be of value.                                              Particular problems to be addressed deal with the scope of                 the support of harmonic measure.  To what extent can it diffuse                 rather than concentrate?  It has been shown recently that such                  measures (on Riemann surfaces) can be absolutely continuous with                respect to area.                                                                     A second class of applications involve the identification of               singular factors in inner functions.  Except for the presence of                omitted values, very few criteria exist which detect these                      factors.  Using new geometric constructs, efforts will be made to               ensure that the resulting functions have prescribed singular                    factors.  It should be noted that Brownian techniques are not                   sensitive enough to detect such terms.
{'FirstName': 'Kenneth', 'LastName': 'Stephenson', 'EmailAddress': 'KENS@MATH.UTK.EDU', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Tennessee Knoxville', 'CityName': 'Knoxville', 'ZipCode': '379163801', 'PhoneNumber': '8659743466', 'StreetAddress': '1331 CIR PARK DR', 'CountryName': 'United States', 'StateName': 'Tennessee', 'StateCode': 'TN'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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