Mathematical Sciences: Overdetermined Systems Defined by Complex Vector Fields

[903007]特里夫斯几个因素促成了在几个复杂变量和偏微分方程的数学领域之间发展起来的密切联系。一个当然是1957年Hans Lewy发现了一个简单的，线性的一阶偏微分方程它没有任何解。这个例子的关键是使用复系数而不是实值系数。在这个项目中进行的部分持续工作属于一个长期的项目来解释和理解这一现象。本研究的背景是表示偏微分方程过定系统的复向量场系统。有人说这样的系统代表微分复合体。复合体的一个代数特征是称为上同调的量。实际上上同调是有层次的；我们感兴趣的是第p次上同调。这是有趣的，因为有强有力的证据表明，p次上同调的消失与相关偏微分方程解的水平集在p-1维上同调的琐碎性有关。计划对这种关系进行系统的研究。第二方面的研究将涉及通过同伦方法的应用对微分方程局部可解性的阻碍。这一领域的最新进展表明，在某些重要的微分算子上，即使下域在边界上不满足弱拟凸条件，也可以使用同伦公式。

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.