Mathematical Sciences: Differential Geometry and Geometric  Analysis
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The investigators will continue previous work on a number of               projects in differential geometry and geometric analysis. Jeff                  Cheeger plans to gain a deeper understanding of invariants                      associated with certain families of elliptic operators which have               recently been of some interest in Theoretical Physics.  He is                   also proposing extensive work toward obtaining further insight                  into the interaction of bounded geometries, collapsing, and                     characteristic numbers for complete Riemanian manifolds.  In                    addition, he will continue to expand his approximation theory for               curvature of smooth manifolds by piecewise linear ones.  Gromoll                expects further substantial progress in his efforts to establish                a global structure theory for complete Riemannian spaces of                     nonnegative Ricci curvature, in particular finiteness results and               the construction of additional classes of examples.  He also                    wants to proceed with his analysis of rigidity phenomena of                     metric foliations, i.e. locally everywhere equidistant                          partitions, in nonnegatively curved manifolds, which has turned                 out to be a crucial and delicate aspect of the global geometry of               such spaces.  In the area of submanifold theory he will work on                 various extensions  of recent global results concerning isometric               deformation of euclidean submanifolds, as well as explore the new               theory of minimal real Kaehler submanifolds, a special but ample                class of particularly interesting minimal submanifolds that                     surfaced in that context.  Ebin plans to work in three areas all                of which involve free boundary value problems.  The first is the                study of the motion of a thin membrane.  The second is the study                of dynamics of a liquid drop, and the third is the liquid drop                  model for a large nucleus.  The second and third areas are both                 studied as examples of dynamical systems constrained to                         submanifolds of function spaces.  A key step in this analysis is                the study of their relative curvature.  Teleman intends to                      continue his work on the Novikov conjecture, an outstanding                     problem in Differential Topology and Operator Theory.  He also                  wants to discover a new approach to Pontrjagin classes of                       combinatorial manifolds, which would also be of significant                     interest in Theoretical Physics.
[{'FirstName': 'Jeff', 'LastName': 'Cheeger', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jeff Cheeger', 'EmailAddress': 'cheeger@cims.nyu.edu', 'NSF_ID': '000172397', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Co-Principal Investigator'}, {'FirstName': 'Detlef', 'LastName': 'Gromoll', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Detlef Gromoll', 'EmailAddress': 'detlef@math.sunysb.edu', 'NSF_ID': '000194162', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}]
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[{'Code': '1265', 'Text': 'GEOMETRIC ANALYSIS'}, {'Code': '1266', 'Text': 'APPLIED MATHEMATICS'}]
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