Mathematical Sciences: Problems in Several Complex Variablesand Partial Differential Equations.

这个项目的工作继续对偏微分方程解与定义在多个复变量空间中的曲面上的映射相关的问题进行数学研究。检验解析性的一般柯西-黎曼方程被表示为一阶微分式。在研究这些表达式时，人们试图确定定义在曲面和区域边界上的函数是否可以作为全纯函数扩展到与曲面相邻的区域。在这些研究中的一个基本工具是将解析圆盘嵌入到高维复数空间中，其边界进入实超曲面或更一般的子流形。这类圆盘映射的微分的消失或不消失，对超曲面映射的研究以及前面提到的扩张问题都有着深远的影响。最终，这项研究试图揭示更多关于流形之间的映射，特别是确定在双全纯变换下保持的新的几何和代数不变量。第二方面的研究来自最近完成的一系列论文，这些论文导致了50年代中期S提出的一个基本可扩性问题的完全解决：泛型流形在一点的极小性是保证在该点附近的每个柯西-黎曼函数全纯扩张到一个楔形的充要条件。即使在简单的情况下，对可扩展性楔形的描述也是未知的。将在寻找可伸缩性楔形的几何描述方面进行工作。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是描述关系，而是从它们中提取定性和定量的意义。

Work on this project continues mathematical research into problems relating solutions of partial differential equations with mappings defined on surfaces in spaces of several complex variables. The general Cauchy-Riemann equations which test for analyticity are expressed as first-order differential expressions. In studying these expressions one tries to decide whether functions defined on surfaces and boundaries of domains can be extended as holomorphic functions to regions adjacent to the surfaces. A fundamental tool in these studies is the embedding of analytic discs into complex spaces of higher dimension with their boundaries going into real hypersurfaces or more general submanifolds. The vanishing or nonvanishing of the differential of such disc mappings has far-reaching implications in the study of mappings of hypersurfaces as well as to the extension problem already mentioned. Ultimately the research seeks to reveal more about mappings between manifolds, in particular to determine new geometric and algebraic invariants preserved under biholomorphic transformations. A second line of investigation follows from the recent completion of a succession of papers leading to the complete resolution of one of the fundamental extendibility questions raised by Lewy in the mid-50's: minimality of a generic manifold at a point is necessary and sufficient to guarantee that every Cauchy-Riemann function near the point extends holomorphically to a wedge. A description of the wedge of extendibility is unknown even in simple cases. Work will be done in finding a geometric description of the wedge of extendibility. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.