Mathematical Sciences: Geometric Function Theory in SeveralComplex Variables

在这个项目中，围绕着几个复变量的理论，将开发三个主要的数学主题。首先，将应用偏微分方程组的技术，特别是实变量微局部技术，深入了解函数的光滑性和包含该曲线的曲线(曲面)类型之间的关系。在一个复数维中，人们要么断定该函数必须是光滑的，要么它的傅立叶系数缓慢衰减。在更高的维度上，这种现象被认为与波前集合的性质有关。相关工作将涉及柯西-黎曼函数的唯一性问题。重点讨论沿非复切向曲线的无限级消失函数的刻画，以及非平坦性假设与CR映射的正则性的相关性。此外，还将研究n维复空间到其自身的全纯映射的基本问题。其中典型的映射是其雅可比是具有有限体积像的单位的映射(它们存在)和自同构，由于未知的原因，它不区分可数稠密集，但在离散可数集上的作用不同。这一研究将应用于对复空间几何和定义在流形上的复偏微分算子的一般理解。

Three main mathematical themes will be developed in this project, built around the theory of several complex variables. In the first, techniques from partial differential equations, especially real variable microlocal techniques, will be applied to provide insight into the relationship between the smoothness of a function and the type of curve (surface) which contains the curve. In one complex dimension one either concludes that the function must be smooth or else its Fourier coefficients decay slowly. In higher dimensions this phenomenon is believed to be related to the nature of the wave front set. Related work will involve questions of uniqueness for Cauchy -Riemann functions. Emphasis will be placed on the characterization of functions vanishing of infinite order along curves which are not complex tangential and on the relevance of the hypothesis of nonflatness as it relates to the regularity of CR-mappings. Additionally, work will be done on fundamental questions concerning holomorphic maps from complex n-dimensional space to itself. Typical of these are maps whose Jacobian is unity having images of finite volume (they exist) and automorphisms which, for reasons unknown, do not distinguish countable dense sets, but act differently on discrete countable sets. This research will have application to the general understanding of the geometry of complex space and to complex partial differential operators defined on manifolds.