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

该项目的工作继续进行数学研究， 偏微分方程解的相关问题 具有定义在多个复空间中的曲面上的映射的 变量 一般的柯西-黎曼方程， 解析性表示为一阶微分 表情 在研究这些表达时，人们试图决定 定义在域的表面和边界上的函数 可以作为全纯函数扩展到邻近的区域， 表面。 这些研究中的一个基本工具是 解析圆到高阶复空间的嵌入 维度，其边界进入真实的超曲面，或 更一般的子流形 的消失或不消失 这种盘映射的微分具有深远的意义 在超曲面映射的研究中， 前面提到的扩展问题。 最终，研究 试图揭示更多关于流形之间的映射，在 特别是确定新的几何和代数不变量 在双全纯变换下保持不变的第二线 调查是根据最近完成的一项继承 的文件，导致完全解决的一个 Lewy在1998年的《自然》中提出了基本的可扩展性问题， 50年代中期：一般流形在一点的极小性是 必要且充分地保证每个柯西-黎曼 函数在点附近全纯地延伸到楔形。 一 对可扩展性楔形的描述甚至在 简单的案子 工作将在寻找一个几何 可扩展性楔形的描述。 偏微分方程是 物理科学中的数学建模。 的现象 包括连续变化，如运动、材料 和能量都服从某些普遍规律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义。

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.