Mathematical Sciences: Harmonic Analysis and Differential   Equations
{'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}
Research on this project will cover broad areas of modern                  mathematical analysis.  Particular topics include harmonic                      analysis, singular integrals, quantum mechanics and several                     complex variables.  Much of the work derives from geometric                     ideas.  Included will be studies of singular Radon transforms                   along submanifolds in which the connection between boundedness of               the transform and curvature properties of the submanifolds will                 be clarified.  Significant progress has been made when the                      underlying structure is a nil-potent Lie group.  Other work                     concerns smooth pseudoconvex domains in the space of several                    complex variables.  Substantial advances have been made in                      understanding the nature of the two basic operators related to                  function theory studied on the domains: the Szego and Bergman                   projections.  Holder estimates have been determined in the two-                 dimensional case.  Further efforts will be made in eliminating                  the dimension requirement and finding sharp estimates for                       functions in Sobolev and Lebesgue spaces.                                            Work is also continuing on a mathematical formulation                      providing a basis for the formation of atoms.  The main issue is                that of estimating the best constant in the stability of matter                 inequality.  In addition, a program to identify all invariant                   polynomials attached to the Taylor expansion of a conformal                     metric at a point of a manifold will be continued.  A complete                  list is believed to have been identified in all odd dimensions,                 but this has yet to have been verified.  Somewhat related is work               on the nodal sets of eigenfunctions of the Laplace operator on                  compact manifolds.  It has been conjectured that these sets have                codimension one Hausdorff measure on the order of the square root               of the corresponding eigenvalue.  This has now been established                 for manifolds with analytic metrics.  Future work will                          concentrate on extending the result to metrics which are                        infinitely differentiable.
{'FirstName': 'Elias', 'LastName': 'Stein', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Elias Stein', 'EmailAddress': 'stein@math.princeton.edu', 'NSF_ID': '000123616', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Princeton University', 'CityName': 'PRINCETON', 'ZipCode': '085442001', 'PhoneNumber': '6092583090', 'StreetAddress': '1 NASSAU HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'New Jersey', 'StateCode': 'NJ', 'CONGRESSDISTRICT': '12', 'CONGRESS_DISTRICT_ORG': 'NJ12', 'ORG_UEI_NUM': 'NJ1YPQXQG7U5', 'ORG_LGL_BUS_NAME': 'THE TRUSTEES OF PRINCETON UNIVERSITY', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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