Mathematical Sciences: Geometry of Singularities
{'Code': '03040200', 'Directorate': {'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Ralph M. Krause'}
In recent years several important advances have been made in               the theory of singularities of complex analytic spaces and                      mappings:  introduction of local polar varieties (by Le Dung                    Trang and Teissier), existence of a canonical Whitney                           stratification (by Teissier), topological characterization of                   Whitney stratifications (by Le Dung Trang and Teissier), Morse                  theory on singular spaces (by Goresky and MacPherson), structure                of the tangent spaces at singular points (by Le Dung Trang and                  Teissier), Hodge theory on singular spaces (by Steenbrink,                      Zucker, Varchenko, Saito), its relation with the study of linear                differential systems via the Riemann-Hilbert correspondence (by                 Mebkhout, Kashiwara, Kawai), triangulation of stable maps                       (Verona).                                                                            These advances give new tools to investigate systematically                basic problems in the theory of singularities.  Gaffney and Le                  will investigate the following problems:                                             (1) The problem of equisingularity of families of mappings,                e.g. the problem of finding algebraic invariants to determine if                a family of mappings is topologically trivial.  The concept was                 introduced by Zariski for spaces and it was shown by Le and                     Ramanujam that the constancy of the Milnor number gives the                     topological equisingularity of a family of complex hypersurfaces.                    (2) The relative  Lefschetz Theorem on hyperplane sections                 for maps from singular spaces into Pn.                                               (3) The canonical decomposition of the local monodromy of                  complex hypersurfaces and its link with the resolution of                       singularities and the geometry of Polars.                                            (4) The conormal structure and higher order conormal                       structure of singular spaces:  this should lead to results on the               geometry of linear differential systems.                                             Furthermore, the techniques involved are quite effective,                  and the investigators hope to be able to use existing software to               implement algorithms to determine the shape and the topology of                 complex analytic singularities.
[{'FirstName': 'Dung Trang', 'LastName': 'Le', 'EmailAddress': None, 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}, {'FirstName': 'Terence', 'LastName': 'Gaffney', 'EmailAddress': 'gaff@neu.edu', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Co-Principal Investigator'}]
{'Name': 'Northeastern University', 'CityName': 'BOSTON', 'ZipCode': '021155005', 'PhoneNumber': '6173733004', 'StreetAddress': '360 HUNTINGTON AVE', 'CountryName': 'United States', 'StateName': 'Massachusetts', 'StateCode': 'MA'}
[{'Code': '1267', 'Text': 'TOPOLOGY'}, {'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}]
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8803478.xml'}