Mathematical Sciences: Partial Differential Equations, GroupRepresentations and Number Theory

埃什普雷斯 8805909 该项目将包括三个数学领域 分析，每一个都代表了早期的延续 research. 第一个问题涉及的解决方案的扩展 偏微分方程的域， 当楔形体与周围域的接触时， 共享共同的边界元素。 在其一般形式中， 是确定柯西数据之间的关系， 边界，这限制了解的可拓性。 解决方案 在低的常系数方程中， 尺寸 工作也将做散射理论的问题。 非交换力学的一个新概念， 空间变量的泊松括号使得空间和动量 分离的李代数（而不是消失），当 适当地配对。 这种方法导致了一种新的概念， 完全可积系统，可应用于 非交换群上的散射问题 工作计划 在更高的维度上发展这项研究。 还将继续努力， 当李群上的左不变算子可以是 在无穷可微函数上的满射 这 这项工作源于早期对所谓的路易的研究 算子-一个没有解的偏微分方程。 使用Husenberg组的初步结果表明， 的满射性的一般框架 微分算子，因此，条件下，一个 可以期望相应的非齐次方程的解 方程

Ehrenpreis 8805909 This project will encompass three areas of mathematical analysis, each of which represents a continuation of earlier research. The first concerns the extensions of solutions of partial differential equations from domains which have the shape of convex wedges to surrounding domains when the wedges share common boundary elements. In its general form the goal is to determine the relations among the Cauchy data on the boundary which imly the extendability of solutions. Solutions have been obtained for constant coefficient equations in low dimentions. Work will also be done on problems of scattering theory. A recent concept of nonabelian mechanics in which the poisson bracket of space variables makes the space and momentum into seperate Lie algebras (rather than vanishing), when suitably paired. This approach leads to a new notion of completely integrable systems which can be applied to scattering problems on nonabelian groups. Work planned seeks to develop this research in higher dimensions. Efforts will also continue on the question of determining when a left-invariant operater on a Lie group can be surjective on the infinitely differentiable functions. This work derives from earlier research on the so-called Lewy operater - a partial differential equation with no solutions. Initial results using the Husenberg group suggest that a general framework for deciding on the surjectivity of differential operators and hence the conditions under which one can expect solutions of the corresponding inhomogeneous equations.