Mathematical Sciences: Geometry of Integrable Systems

8901607 Sattinger这个项目涉及基于半单李代数的哈密顿层次结构的研究，半单李代数被视为平坦连接的规范理论。平面连接出现在许多问题中，如自对偶杨-米尔斯方程和微分几何中的嵌入问题。在求解直线上n×n一阶系统逆散射问题的基础上，提出了一种将Sato的Kadomtsev-Petviashvili系统的无穷维格拉斯曼图推广到n×n系统的方法。从规范变换和Riemann-Hilbert分解的观点出发，研究了n×n系统逆散射问题的伴随作用。这种方法是基于散射理论的主丛形式的结果的表述，以及Zakharov和Shabat的“穿衣方法”。

8901607 Sattinger This project concerns the investigation of Hamiltonian hierachies based on semi-simple Lie algebras viewed as gauge theories of a flat connection. Flat connections arise in a multitude of problems, such as the self-dual Yang-Mills equations, and embedding problems in differential geometry. An approach to extending Sato's infinite dimensional Grassmannian picture of the Kadomtsev-Petviashvili hierachy to the nxn hierarchies is proposed, based on the solution of the inverse scattering problem for nxn first order systems on the line. The co-adjoint action for the hierachies is to be investigated from the point of view gauge transformations and the Riemann-Hilbert factorization associated with the solution of the inverse scattering problem for nxn systems. The approach is based on the formulation of the results of scattering theory in terms of principal bundles, and the "dressing method" of Zakharov and Shabat.