Mathematical Sciences: Cyclic Cohomology, Symbols, Local    Index, Toeplitz-type Operators and Hyponormal Operators
{'Code': '03040103', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': None, 'PO_EMAI': None, 'PO_PHON': None}
Operator theory is a central discipline in Modern                         Analysis.  Its origins lie in the study of mathematical physics                 and partial differential equations in the early twentieth                       century.  At that time, it was seen that numerous physical                      problems in the theory of equilibria, vibration, quantum                        mechanics, etc. could be studied productively via the integral                  equations that model the phenomena.  Currently, operator theory                 has a strong interaction not only with physics, but also                        engineering and many branches of mathematics.  Notable in the                   latter regard are K-theory, non-commutative differential                        geometry, index theory, and cohomology.                                               Professor Xia is an expert in all of these aspects of                     operator theory, with deep and significant contributions to his                 credit.  For many years, before taking residency in the United                  States, he was an active, prolific leader in Chinese mathematics.               He has done outstanding work in function theory, quasi-invariant                measures on infinite-dimensional spaces, and topological                        algebras.  He has worked on scattering in quantum field theory,                 and in harmonic analysis he has collaborated with the great                     Soviet mathematician Gelfand.  In recent years, he has                          concentrated on operator theory.  This work has involved deep                   connections among hyponormal operators, contractions, and                       K-theory; cyclic cohomology and almost Lie algebras of operators;               almost Lie groups of operators; and difference operators and                    mathematical physics.                                                                     The proposed research focuses on three major areas of                 research.  The first involves determining cyclic cohomology and                 relations among cocycles, symbols, and index.  This relates to K-               theory, non-commutative differential geometry, and quantum field                theory.  The second is the study of analytic models of hyponormal               operators and subnormal operators, with relationship to invariant               subspaces, cohomology, and algebraic geometry.  Finally,                        Professor Xia will study Toeplitz operators and applications to                 differential equations and pseudo-differential operators.
{'FirstName': 'Daoxing', 'LastName': 'Xia', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Daoxing Xia', 'EmailAddress': 'daoxing.xia@vanderbilt.edu', 'NSF_ID': '000460847', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Vanderbilt University', 'CityName': 'NASHVILLE', 'ZipCode': '372032416', 'PhoneNumber': '6153222631', 'StreetAddress': '110 21ST AVE S', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Tennessee', 'StateCode': 'TN', 'CONGRESSDISTRICT': '05', 'CONGRESS_DISTRICT_ORG': 'TN05', 'ORG_UEI_NUM': 'GTNBNWXJ12D5', 'ORG_LGL_BUS_NAME': 'VANDERBILT UNIVERSITY', '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': '1262', 'Text': 'MODERN ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8700048.xml'}