Mathematical Sciences: Singular Integrals and Fourier       Integrals
{'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}
The mathematical theme of this research is a combination                   of harmonic analysis and the modern theory of differential                      operators applied to questions about integral transforms of                     certain type.  A physical counterpart to this analysis occurs                   in x-ray reconstruction or tomography where one seeks to                        visualize an unknown object by information derived from lower                   dimensional pictures taken from a limited number of angles.                          In the mathematical setting, it is assumed that a class of                 surface averages (Radon transformations) are known, where the                   surface here may be an abstract manifold, and form what is                      called a Fourier integral operator.  Efforts will be made to                    obtain estimates for such averages or transforms in what is                     called the singular case.  Examples in this particular context                  occur naturally when one considers Hilbert transforms along                     curves or operators associated with the d-bar Neumann problem                   on weakly pseudoconvex domains in several complex variables.                    When the transforms are not degenerate, mean-square estimates                   can be extended to all power norms.  This work will concentrate                 on the degenerate case where estimates are harder to obtain                     and, in fact, the norm estimates for Radon transforms may fail,                 except in the mean-square case.                                                      Work will also be done in an attempt to formulate classes                  of Fourier integral operators sufficient to deal with the                       problem of admissibility and invertibility for the restricted                   k-plane transform in Euclidean space.  The construction of the                  relevant canonical relations should be related to the                           Grassmannian geometry (of the k-planes).
{'FirstName': 'Allan', 'LastName': 'Greenleaf', 'PI_MID_INIT': 'T', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Allan T Greenleaf', 'EmailAddress': 'allan@math.rochester.edu', 'NSF_ID': '000085886', 'StartDate': '07/01/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Rochester', 'CityName': 'ROCHESTER', 'ZipCode': '146113847', 'PhoneNumber': '5852754031', 'StreetAddress': '910 GENESEE ST', 'StreetAddress2': 'STE 200', 'CountryName': 'United States', 'StateName': 'New York', 'StateCode': 'NY', 'CONGRESSDISTRICT': '25', 'CONGRESS_DISTRICT_ORG': 'NY25', 'ORG_UEI_NUM': 'F27KDXZMF9Y8', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF ROCHESTER', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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