Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

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9401020 Kappeler This award supports mathematical research focusing on infinite systems of differential equations known as Hamiltonian systems. They derive from models of physical systems and nonlinear partial differential equations obtained through variational principles. Work to be done includes the analysis of the symplectic structure of the phase space of completely integrable Hamiltonian systems to help understand Hamiltonian perturbations. Studies of regularized determinants of elliptic operators will also be carried out. They appear in modern physics (functional integrals) as well as in geometry and topology (torsion and eta-invariants). This work continues efforts to extend results on asymptotic expansions of the logarithm of the determinant of one-parameter families of pseudodifferential operators. A third line of investigation concerns the investigation of microlocal smoothing properties for a large class of linear and nonlinear systems of dispersive evolution equations of Schrodinger type. Finally, work will bedone on spectral problems and related inverse problems for Schrodinger operators on line bundles over tori and Heisenberg manifolds. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***