Several Complex Variables and Applications

摘要奖：DMS-9971628首席研究员：Laszlo Lempert首席研究员将在两个主题上工作。受到几何和物理例子的启发，研究者将研究无限维复流形，特别是这种流形上的非齐次柯西-黎曼方程，这是一个与一系列几何和分析问题有关的微分方程组。如果该程序成功，它可能导致关于复流形的定义的Newlander-Nirenberg定理的推广。在无限维环境中处理复杂分析的方法非常少，而本研究的目的之一将是发明必要的工具。另一个主题是柯西-黎曼流形的可嵌入性。这项研究将集中在紧致的、强伪凸的3维柯西-黎曼流形上，这是本课题研究的最困难的维度。我们将研究Cauchy-Riemann函数的空间大小如何受到流形本身的几何或流形所覆盖的复流形和几乎复流形的几何的影响。另一个方面与球上的离散群作用有关。这些工具来自椭圆型偏微分方程、几何学和可能的代数几何。这项研究的主要主题是几个，可能是无穷多个变量的函数。这类函数出现在研究可由许多参数描述的复杂系统时。例如，太空探索岩石的位置用三个数字来描述：坐标；火箭可能会旋转，在这种情况下，它旋转的速度也是参数，等等。其他量可由这些参数确定，例如，到达火箭所需的无线电信号强度：它们是参数的函数。除了上述所谓的内部参数外，系统构建后还会确定一些外部参数，例如重量。这项研究调查了某些函数如何随着参数(内部或外部)的微小变化而变化；也研究了理解“微小的”或甚至是微小的变化如何有助于理解大的变化。这个项目的两个细节值得一提的是。我们将专注于复杂的参数而不是实数-尽管这一假设具有违反直觉的性质，但它具有实用价值。此外，此项目的一部分处理无限多个参数的函数。自然界中是否存在这样的系统还有待讨论，但毫无疑问，它们代表着具有大量参数的系统的近似值，这些参数确实存在。

AbstractAward: DMS-9971628Principal Investigator: Laszlo LempertThe principal investigator will work on two topics. Motivated byexamples from geometry and physics, the investigator will studyinfinite dimensional complex manifolds, especially theinhomogeneous Cauchy--Riemann equations on such manifolds, asystem of differential equations relevant to a host of geometricaland analytical problems. If the program is successful, it maylead to extensions of the Newlander--Nirenberg theorem concerningthe very definition of complex manifolds. There are very fewmethods to deal with complex analysis in an infinite dimensionalsetting and one of the aims of this research will be to invent thenecessary tools. The other topic concerns embeddability ofCauchy--Riemann manifolds. This research will focus on compact,strongly pseudoconvex Cauchy--Riemann manifolds of dimension 3,which is the hardest dimension for the subject. We will study howthe size of the space of Cauchy--Riemann functions is influencedby the geometry of the manifold itself, or by the geometry ofcomplex and almost complex manifolds spanned by the manifold.Another aspect is related to discrete group actions on balls. Thetools are from elliptic partial differential equations, geometryand possibly algebraic geometry.The main subjects of this research are functions of several,possibly infinitely many variables. Such functions arise in thestudy of complicated systems that can be described by a number ofparameters. For example the position of a space exploring rocketis described by three numbers, the coordinates; the rocket mightbe spinning, in which case the speed by which it spins is also aparameter, and so on. Other quantities may be determined by theseparameters, e.g. the necessary intensity of a radio signal toreach the rocket: they are functions of the parameters. Inaddition to the above so called internal parameters, there arealso external parameters determined once the system isconstructed, e.g. its weight. This research investigates howcertain functions vary as parameters (internal, or external) areallowed to change a little; and also how understanding "little" oreven infinitesimal variations helps in understanding largevariations. Two specifics of this project are worth mentioning. Wewill focus on parameters that are complex rather than realnumbers---despite its counterintuitive nature, this assumption haspractical value. Also, part of this project treats functions ofinfinitely many parameters. Whether such systems exist in natureis up for debate, but undoubtedly they represent approximations tosystems with a huge number of parameters, which do exist.