Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

9703847 T.Kappeler 1.无限维哈密顿系统：对有限维完全可积哈密顿系统相空间的辛结构的分析在哈密顿摄动的研究中是非常有用的。研究者计划将先前关于KdV方程(KdV)作用角度变量的工作应用于KdV或KdV族中任何方程的哈密顿摄动(Kam型定理)的解析、准周期(时间)解的存在性问题的研究，并研究摄动KdV方程(Nekhoroshev型估计)解的长期稳定性。此外，研究者计划在一般情况下证明一大类无限维完全可积的哈密顿系统的(广义)作用角变量的局部存在性。2.椭圆算子的正则化行列式：它们出现在现代物理(泛函积分)以及几何学和拓扑学(挠率，ETA不变量)中。研究人员计划继续他的工作，以分析正则行列式的技术及其在几何和拓扑学中的应用：正则行列式的扭转和相对扭转(在通常和L2背景下)；通过形变的正则化行列式的数值计算；行列式类的流形和闭流形M的L2扭转的表示，作为与M的有限覆盖序列相关的适当正规化扭转序列的极限。3.色散波的平滑：在许多情况下，例如在水波的研究中出现色散波。研究者计划分析一大类薛定谔类型的色散发展方程的线性和非线性系统的微局部光滑性。在许多不同的应用中，如信号沿光纤的传输(电信)，水波和水流的分析，以及理论物理(天体力学，量子场论)，基本的理想模型变成了有限或无限维的可积系统。为了便于应用，必须研究这些理想模型的扰动。虽然有限维的可积系统及其扰动已被较好地理解，但在无限维的情况下仍有许多工作要做。

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.