Mathematical Sciences: Complex and Harmonic Analysis
{'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 focus on recent developments in the theory of                 quasiconformal groups, the universal Teichmuller space and                      domains arising in approximation theory termed John domains.                    Some classical questions regarding quasiconformal mappings will                 also be reexamined in light of new results obtained by several                  researchers.  Among these are questions about the approximation                 of quasiconformal maps in higher dimensions by differentiable                   maps of the same dilatation.  Efforts to characterize the                       quasiconformal groups (discrete ones, first) have energized                     several top investigators including the principal investigator.                 The fundamental question is to determine when a group is                        effectively a Mobius group.  Since this is not always the case,                 the problem of filling in the middle should generate some                       exciting research.                                                                      Other work oriented more toward functional-analytical                   structures in complex analysis will focus on the study of cyclic                vectors in spaces of analytic functions.  These are elements of                 the spaces whose powers span the space.  General conditions for                 cyclicity are only known for special spaces.  However, for Bloch                spaces only partial results are available and work will continue                to complete the picture here.                                                           In addition, investigations will be carried out on                      injectivity criteria for local homeomorphisms, quasisymmetric                   groups and Markov processes.  Analysis in the field of several                  complex variables will focus on outer functions in the polydisk                 and ball in two complex variables, especially when the zero set                 is small.  An old problem of B. Levin on integrable functions on                the quarter plane will be pursued.                                                      This work relates to a number of important areas of                     mathematics including three-manifold theory, partial differential               operators and Toeplitz transformation spaces of analytic                        functions. It is also used in engineering applications involving                systems theory.
{'FirstName': 'Frederick', 'LastName': 'Gehring', 'PI_MID_INIT': 'W', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Frederick W Gehring', 'EmailAddress': 'fgehring@umich.edu', 'NSF_ID': '000325210', 'StartDate': '06/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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