Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

9401020 Kappeler该奖项支持专注于被称为哈密顿系统的无限微分方程组的数学研究。它们来源于物理系统模型和通过变分原理得到的非线性偏微分方程式。要做的工作包括分析完全可积哈密顿系统相空间的辛结构，以帮助理解哈密顿扰动。此外，还将研究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(挠率和ETA不变量)中。本文继续推广关于单参数拟微分算子族行列式的对数的渐近展开式的结果。第三个研究方向是研究一大类薛定谔型色散发展方程的线性和非线性系统的微局部光滑性。最后，我们将研究Tori流形和Heisenberg流形上线丛上薛定谔算子的谱问题和相关的逆问题。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***