Several Complex Variables and Applications

摘要奖：DMS-9971628首席研究员：Laszlo Lempert首席研究员将研究两个主题。 受几何和物理实例的启发，研究者将研究无穷维复流形，特别是此类流形上的非齐次Cauchy-Riemann方程，与许多几何和分析问题相关的微分方程系统。 如果该计划是成功的，它可能会导致扩展的纽兰德-尼伦伯格定理有关的复杂流形的定义。 在无限维环境下处理复分析的方法很少，本研究的目的之一将是发明必要的工具。 另一个主题是关于Cauchy-Riemann流形的可嵌入性。 本研究将集中在紧的，强伪凸的三维柯西-黎曼流形，这是最难的维度的主题。 我们将研究Cauchy-Riemann函数空间的大小如何受到流形本身几何形状的影响，或者受到流形所张成的复流形和几乎复流形几何形状的影响。另一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学。本研究的主要对象是多个，可能是无穷多个变量的函数。这些函数出现在研究复杂系统的过程中，这些复杂系统可以用许多参数来描述。例如，空间探测火箭的位置可以用三个数字来描述：坐标;火箭可能在旋转，在这种情况下，它旋转的速度也是近似的，等等。其他的量可以由这些参数来确定，例如，到达火箭的无线电信号的必要强度：它们是这些参数的函数。除了上述所谓的内部参数外，还有一些外部参数是在系统构建后确定的，例如其重量。这项研究调查了当参数（内部或外部）允许改变一点时，某些函数如何变化;以及理解“小”或甚至无穷小的变化如何有助于理解大变化。该项目的两个具体情况值得一提。我们将关注复杂的参数，而不是实数-尽管它违反直觉的性质，这个假设有实际价值。 此外，该项目的一部分将处理大量参数的函数。 这样的系统是否存在于自然界还有待讨论，但毫无疑问，它们代表了具有大量参数的系统的近似值，而这些参数确实存在。

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.