9703847 T. Kappeler 1. Hamiltonian systems of infinite dimension: The analysis of the symplectic structure of the phase space of completely integrable Hamiltonian systems of (in)finite dimension is very useful in the study of Hamiltonian perturbations. The investigator plans to apply prior work on action-angle variables for the Korteweg-deVries equation (KdV) to contribute to the investigation of the problem of the existence of analytic, quasiperiodic (in time) solutions of Hamiltonian perturbations of KdV or of any of the equations in the KdV hierarchy (KAM type theorem) and to study longtime stability of solutions of perturbed KdV equations (Nekhoroshev-type estimates). Further, the investigator plans to prove, in a generic situation, local existence of (generalized) action-angle variables for a large class of completely integrable Hamiltonian systems of infinite dimension. 2. Regularized determinants of elliptic operators: They appear in modern physics (functional integrals) as well as in geometry and topology (torsion, eta-invariant). The investigator plans to continue his work on techniques for analyzing regularized determinants and their applications to geometry and topology: Torsions for bordism and relative torsion (in the usual and the L2-setting); numerical computations of regularized determinants via a deformation; manifolds of determinant class and representation of the L2-torsion of a closed manifold M as a limit of a sequence of appropriately normalized torsions associated to a sequence of finite covers of M. 3. Smoothing of dispersive waves: Dispersive waves appear in many instances, e.g. in the study of water waves. The investigator plans to analyze microlocal smoothing properties for a large class of linear and nonlinear systems of dispersive evolution equations of Schroedinger type. In many different applications, such as propagation of signals along optical fibers (telecommunication), analysis of water waves and currents, and theoretical physics (celestial mechanics, quantum field theory), the basic underlying ideal models turn out to be integrable systems of finite or infinite dimension. To be useful for applications, perturbations of these ideal models have to be studied. Whereas integrable systems of finite dimension and their perturbations are relatively well understood, much remains to be done in the infinite dimensional case.
