Mathematical Sciences: Hamiltonian Systems of Infinite Dimensions

小行星9401020 该奖项支持数学研究，重点是被称为汉密尔顿系统的微分方程的无限系统。 它们来自物理系统的模型和通过变分原理得到的非线性偏微分方程。 要做的工作包括完全可积的哈密顿系统的相空间的辛结构的分析，以帮助理解哈密顿扰动。 还将研究椭圆算子的正则化行列式。 它们出现在现代物理学（泛函积分）以及几何学和拓扑学（挠率和η-不变量）中。 这项工作继续努力，扩大结果的渐近展开的行列式的单参数家庭的伪微分算子。 第三条调查线涉及调查的微观局部光滑性质的一大类线性和非线性系统的色散发展方程的薛定谔型。 最后，我们将研究环面和Heisenberg流形上线丛上Schrodinger算子的谱问题和相关的反问题。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***