Mathematical Sciences: Overdetermined Systems Defined by Complex Vector Fields

8903007有几个因素促成了几个复变量的数学场和偏微分方程之间的密切联系。可以肯定的是，汉斯·路易在1957年发现了一个简单的一阶线性偏微分方程式，它没有任何解。这个例子的关键是使用复数而不是实数系数。该项目开展的部分持续工作属于解释和理解这一现象的长期项目。这项研究的背景是代表超定偏微分方程组的复矢量场系统。有人说，这样的系统代表了一种微分复合体。复数的一个代数特征是一个称为上同调的量。实际上，上同调是有层次的；，这里感兴趣的是第p个上同调。这是因为有强有力的证据表明，p阶上同调的消失与相关偏微分方程解的水平集在p-1维的同调的平凡有关。计划对这种关系进行系统的研究。第二个研究方向是通过应用同伦方法来研究微分方程局部可解性的障碍。这一领域的最新进展表明，即使基本区域不满足边界上的弱伪凸性条件，同伦公式也可以用于某些重要的微分算子。

8903007 Treves Several factors contribute to the close connections which have evolved between the mathematical fields of several complex variables and partial differential equations. One certainly, was the discovery by Hans Lewy in 1957 of a simple, linear partial differential equation of first order which had no solutions whatsoever. The key to the example was the use of complex rather than real-valued coefficients. Part of the continuing work carried out in this project belongs to a long-term project to explain and understand this phenomenon. The context of the research is that of systems of complex vector fields representing overdetermined systems of partial differential equations. One says that such a system represents a differential complex. An algebraic characteristic of complexes is a quantity called the cohomology. Actually there are levels of cohomology; the one of interest here is the p-th cohomology. It is of interest because there is strong evidence to suggest that the vanishing of the p-th cohomology is related to the triviality of the homology, in dimension p-1, of the level sets of the solutions to the associated partial differential equations. A systematic study of this relationship is planned. A second line of investigation will be concerned with obstructions to local solvability of differential equations through the application of homotopy methods. Recent advances in this area indicate that homotopy formulas can be used on certain important differential operators even when the underlying domains do not satisfy even weak pseudo-convexity conditions on the boundary.