Mathematical Sciences: Fourier Analysis and Partial         Differential Equations
{'Code': '03040102', 'Directorate': {'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff'}
Three major areas will be addressed during the course of                   this mathematical research; geometry, analysis and economic                     modeling.                                                                            The Yamabe problem of Riemannian geometry asks if a manifold               with a given Riemannian metric is conformally equivalent to                     another with constant scalar curvature if the new metric is                     obtained from the old by multiplication by a positive function.                 Work will be done on the analogous problem for Cauchy-Riemann                   structures.  Here one seeks to choose a Levi form with constant                 pseudohermitian scalar curvature.  This amounts to solving an                   explicit partial differential equation on the manifold.  Results                on compact strictly pseudoconvex orientable manifolds have been                 extensive for odd dimensions.  Work will proceed on the remaining               cases.  Efforts will also be made to extend fundamental (P, q1) -               estimates between power integrals of a function and the image of                the function under the heat operator.  Results over the past two                decades, while sharp, have rested on the assumption that the two                exponents are in duality.  Such results all fall within the                     general definition of Sobolev inequalities.  The renewed interest               in such comparisons is related to work on uniqueness properties                 of solutions of partial differential equations.  In addition to                 seeking new Sobolev inequalities work will also be done in                      expanding their applicability to more general differential                      operators.                                                                           Recent interest in economics derives from a result of                      Frobenius about differential equations which, in modern                         terminology, states when a one-form has an integrating factor.                  In microeconomics one considers one-forms of differences between                differentials of income and demand functions of income and prices               (multiplied by differentials of prices).  These forms distinguish               whether or not a tangent vector is pointing in the direction of                 improvement.  The Frobenius result is the statement that a                      utility function exists -  it is generally regarded as a basic                  axiom.  There is evidence to suggest that the axiom fails for                   collections of consumers.  The present project will focus on                    characterizing conditions when approximate integrability can be                 expected.
{'FirstName': 'David', 'LastName': 'Jerison', 'EmailAddress': 'jerison@math.mit.edu', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Massachusetts Institute of Technology', 'CityName': 'Cambridge', 'ZipCode': '021394301', 'PhoneNumber': '6172531000', 'StreetAddress': '77 MASSACHUSETTS AVE', 'CountryName': 'United States', 'StateName': 'Massachusetts', 'StateCode': 'MA'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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