Mathematical Sciences: Linear Partial 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}
Work will be done on problems arising in the linear                        theory of partial differential equations.  The research                         involves continued efforts to analyze boundary problems for                     wave equations.  Current investigations focus on classes of                     operators, known as Airy operators, which describe                              reflections of waves by boundaries in the presence of grazing                   or gliding rays.  This work has numerous applications to                        problems in scattering theory.  Studies have focused on a                       symbol calculus for the Airy operators which arise when                         hyperbolic equations reduce to boundary problems via a                          parametrix construction.  The operators are considerably                        different if grazing rays or gliding rays are present.                          Despite these differences a good operator calculus and even a                   useful symbol calculus is developing which will be sufficient                   to treat natural boundary problems in the presence of rays.                          A second area of emphasis in this project concerns                         various ways in which noncommutative harmonic analysis                          applies to problems in partial differential equations.  Two                     particular directions of research are the spectral theory of                    Schroedinger operators associated with gauge fields and                         noncommutative microlocal analysis.  Specific applications to                   chaotic behavior of classical dynamical systems are possible                    through knowledge of the spectral behavior of Schroedinger                      operators.  Tools necessary to treat these problems include                     geometric invariant theory and group representations.                                Other work will concentrate on the application of                          noncommutative microlocal analysis in the study of hyperbolic                   equations with double characteristics, a more difficult                         subject than the hypoelliptic case.  A calculus of Fourier                      integral operators will be developed using noncommutative                       harmonic analysis to extend recent work on pseudodifferential                   operators.  The Heisenberg group has been central to much of                    the current research in this area.  There is a wealth of                        other Lie groups which must be tried out for versions of                        noncommutative microlocal analysis.  This line of research                      will undoubtedly be developed in the coming years.                                   Numerous applications of this work to diffraction of                       waves, hyperbolic systems with viscosity and propagation of                     singularities can be expected.
{'FirstName': 'Michael', 'LastName': 'Taylor', 'PI_MID_INIT': 'E', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Michael E Taylor', 'EmailAddress': 'met@math.unc.edu', 'NSF_ID': '000254452', 'StartDate': '05/15/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of North Carolina at Chapel Hill', 'CityName': 'CHAPEL HILL', 'ZipCode': '275995023', 'PhoneNumber': '9199663411', 'StreetAddress': '104 AIRPORT DR STE 2200', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'North Carolina', 'StateCode': 'NC', 'CONGRESSDISTRICT': '04', 'CONGRESS_DISTRICT_ORG': 'NC04', 'ORG_UEI_NUM': 'D3LHU66KBLD5', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL', 'ORG_PRNT_UEI_NUM': 'N8DMK1K4C2K5'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8801457.xml'}