Mathematical Sciences: Symmetry Problems in Complex Analysisand Potential Theory
{'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}
A two-year award is recommended in support of mathematical              research focusing on approximation in the complex domain and                    associated higher dimensional generalizations.  The principal                   investigator plans to use techniques from functional analysis,                  complex function theory and potential theory to address                         fundamental questions in two areas.  The first concerns the                     correspondence between geometric properties of bounded sets and                 basic approximations related to the sets.  Specifically, the                    principal investigator has established a lower bound for the                    distance between the classical conjugate of the identity function               and the algebra of rational functions on a set.  It is twice the                area divided by the perimeter.  Equality holds for discs and                    annuli.  Work will be done investigating other possible cases of                equality.  Interestingly, this result includes the classical                    isoperimetric inequality.  The solution to this geometric                       question involves questions about the Neumann problem for the                   Laplace operator and ordinary differential equations in the                     complex domain.                                                                         All the concepts for approximating the conjugate identity in            the plane extend to higher dimensions.  This leads to the second                line of study, generalizing classical Schwarz functions to                      dimension greater than two, replacing the conjugate function by                 one which becomes constant under the Laplacian (the distance                    function).  Rational functions go over to harmonic functions (the               closure of the kernel of the Laplacian).  One has an easy upper                 bound on the approximating distance in terms  of the volume of                  the set in question.  The conjectured lower bound is volume                     divided by boundary area.  Results in this context will be harder               to achieve.  However a viable form of the Schwarz function has                  been proposed for higher dimension.  Taken together with                        additional work on the interplay between geometry and basic                     concepts of potential theory, one expects progress toward a                     better understanding of the nature of the lower bound sought.                           This work is expected to have broad application to                      mathematical analysis and the geometry of function spaces.
{'FirstName': 'Dmitry', 'LastName': 'Khavinson', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Dmitry Khavinson', 'EmailAddress': 'dkhavins@usf.edu', 'NSF_ID': '000103194', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Arkansas', 'CityName': 'FAYETTEVILLE', 'ZipCode': '727013124', 'PhoneNumber': '4795753845', 'StreetAddress': '1125 W MAPLE ST STE 316', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Arkansas', 'StateCode': 'AR', 'CONGRESSDISTRICT': '03', 'CONGRESS_DISTRICT_ORG': 'AR03', 'ORG_UEI_NUM': 'MECEHTM8DB17', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF ARKANSAS', 'ORG_PRNT_UEI_NUM': None}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1986&All=true', 'xml': '8618755.xml'}