Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

9703847 T. Kappeler 1。无限维哈密顿系统：分析有限维完全可积哈密顿系统相空间的辛结构对研究哈密顿摄动是非常有用的。研究者计划将先前关于Korteweg-deVries方程（KdV）的作用角变量的工作应用于KdV的哈密顿摄动或KdV层次中的任何方程（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方程解的长期稳定性（nekhoroshevtype估计）。进一步，研究者计划证明，在一般情况下，（广义）作用角变量的局部存在的一大类的无限维的完全可积哈密顿系统。2. 椭圆算子的正则行列式：它们出现在现代物理学（泛函积分）以及几何和拓扑（扭转，不变）中。研究者计划继续他的工作，分析正则行列式及其在几何和拓扑中的应用技术：bordism和相对扭转（在通常和l2设置中）的扭转；变形正则化行列式的数值计算一个封闭流形M的l2 -扭转作为与M的有限覆盖序列相关联的适当归一化扭转序列的极限的表示。色散波的平滑：色散波在许多情况下都会出现，例如在水波的研究中。研究者计划分析一大类薛定谔型色散演化方程的线性和非线性系统的微局部平滑特性。在许多不同的应用中，例如沿光纤传播的信号（电信），水波和水流的分析，以及理论物理（天体力学，量子场论），基本的潜在理想模型被证明是有限或无限维的可积系统。为了对应用有用，必须研究这些理想模型的摄动。有限维的可积系统和它们的微扰相对来说已经被很好地理解了，但是在无限维的情况下还有很多工作要做。

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.