Mathematical Sciences: Research in Global Analysis

三位研究者将研究有限维流形的几何和拓扑，以及在研究这些流形上的常微分方程和偏微分方程时出现的各种函数空间的解析、代数和几何性质之间的相互关系。一位研究者将研究变分问题的对称群及其相关截面和切片的几何。第二章将研究可积系统解的格拉斯曼结构和这些结构的相关painlelevel分析。第三章将分析可积系统的洛朗解与其不变环面几何之间的关系。这三位研究者将扩展某些可积系统的几何和代数理论。这种系统的一个例子是模拟运动物体的物理行为的哈密顿微分方程。它们以一种优雅的方式结合了能量和动量守恒。这些系统的解倾向于循环，不具有包含摩擦的系统的许多特性。

Three investigators will study problems interrelating the geometry and topology of finite dimensional manifolds with analytic, algebraic, and geometric properties of various function spaces that arise in the study of ordinary and partial differential equations on these manifolds. One investigator will study symmetry groups of a variational problem and the geometry of their associated sections and slices. The second will investigate the Grassmannian structures of the solution to integrable systems and the related Painleve analysis of these structures. And the third will analyze the relationship between the Laurent solutions to integrable systems and the geometry of their invariant tori. These three investigators will extend theories involving the geometry and algebra of certain integrable systems. One example of such systems are Hamiltonian differential equations which model the physical behavior of moving objects. These incorporate conservation of energy and momentum in an elegant manner. Solutions to these systems tend to circulate and don't have many of the properties of systems which incorporate friction.