Mathematical Sciences: Smoothing of Dispersive Waves

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The primary area of research supported by this award is partial differential equations, especially those used to model wave phenomena. Work includes the investigation of smoothing properties for a large class of linear and nonlinear dispersive evolution equations. Dispersive waves are characterized by the property that the group velocity depends in a nontrivial way on the wave number. An example would be Korteweg deVries equation. The smoothing phenomenon refers to the observed behavior of solutions of certain equations - they are smoother than their initial conditions. The approach to this work is the use of the commutator method together with scattering theory. One needs microlocal versions of Sobolev weighted norms to replace the standard norms which do not persist with time. Other research directions include regularized determinants: the study of variational formulas for the regularized determinants of elliptic operators on Riemannian manifolds of dimension two; global aspects of completely integrable Hamiltonian systems and the Schrodinger operator on tori and Heisenberg manifolds. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.