Mathematical Sciences: Differential Equations, Probability, and Applications
{'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}
Broad areas of mathematical analysis will be covered by work               to be done.  The central theme is partial differential equations,               both theory and applications.  In addressing a variety of a                     fundamental questions techniques will be employed from                          probability, topology, calculus of variation and geometry.  The                 equations arise from diverse sources such as mathematical                       physics, liquid crystals, nerve conduction, elasticity and fluid                dynamics.                                                                            Among the specific topics addressed are questions concerning               fully nonlinear elliptic equations.  Typical of these is the                    Monge-Ampere equation.  In contrast to classical partial                        differential equations, the fully nonlinear ones are not solvable               in all domains.  To date, strictly convex domains have provided                 satisfactory solutions, but a determination of natural conditions               on domains for solvability is not yet on the horizon.  Recent                   progress in the use of variational methods on nonlinear problems                has produced important information regarding the existence of                   time-periodic solutions in vibration problems.  Using index                     concepts it has been possible to determine multiple solutions of                minimal period.  The index concept will be studied further in the               context of general nonlinear wave equations to find analogous                   period solutions of minimal period.                                                  The appearance of theta functions in the theory of                         integrable Hamiltonian systems of differential equations such as                the Toda lattice and Korteweg - deVries equations suggests that                 analogous functions and geometry lies behind most of the                        classical infinite systems.  Efforts in several specific cases                  have borne fruit in the establishment of a complete apparatus of                curves, divisors, differentials, Abel sums etc. in each case.                   Further work will continue along these lines.                                        Other work will focus on similarity of nonlinear flows for                 partial differential equations, on the failure of scattering in                 Burger's equation and on resonance of continuous spectra.  The                  conclusions of Hartman's major result on similarity of solutions                of ordinary differential equations to solutions of linear                       equations do not apply when one goes over to infinite systems,                  leaving many important questions unanswered at this time.
{'FirstName': 'Louis', 'LastName': 'Nirenberg', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Louis Nirenberg', 'EmailAddress': 'nirenbrg@acf4.nyu.edu', 'PI_PHON': '2129983192', 'NSF_ID': '000159874', 'StartDate': '07/15/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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