Mathematical Sciences: Existence and Multiplicity Questions for Periodic Solutions of Hamiltonian Systems and Related   Topics
{'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}
The theme of this work centers on the structure of solutions               of systems of ordinary differential equations.  The equations                   have roots in classical mechanics described by Hamiltonian                      systems.  To understand the orbit structures of these systems,                  one tries to find invariant subsystems, the simplest of those                   being periodic solutions.  This leads naturally to boundary value               problems.                                                                            Three different questions will be treated in this project.                 The first is concerned with conditions one can expect to find                   periodic solutions on a prescribed energy surface.  Complete                    results are available for convex and star-shaped surfaces.  Work                will be done in expanding known theory to compact hypersurfaces                 of contact type.  The expected criterion for periodic solutions                 is the vanishing of the first (real) homology group.  A newly                   observed phenomenon of almost existence of periodic solutions                   will also be pursued.                                                                A second direction this work will follow is one of finding                 symplectic invariants for periodic solutions on compact                         hypersurfaces in symplectic manifolds and deciding on                           multiplicity questions.  In attempting to find the number of                    periodic solutions on an energy surface, one is faced with the                  difficulty that the usual variational approach does not give the                unparametrized periodic solutions.  For compact convex surfaces,                efforts will be made to establish this number - which is thought                to be at least half the dimension of the ambient space.                              The third area of concentration will deal with symplectic                  geometry.  Considerable activity has been stimulated by the                     solution of one of Arnold's conjectures and the introduction of                 holomorphic maps of Gromov.  Current work will focus on the role                of the 2nd homotopy group in the Lagrangian intersection problem.
{'FirstName': 'Helmut', 'LastName': 'Hofer', 'PI_MID_INIT': 'H', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Helmut H Hofer', 'EmailAddress': 'hofer@cims.nyu.edu', 'NSF_ID': '000197506', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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