Mathematical Sciences: Geometric Function Theory in SeveralComplex Variables

三个主要的数学主题将在这个项目中发展，建立在几个复杂变量的理论。首先，偏微分方程的技术，特别是实变量微局部技术，将被应用于了解函数的平滑度与包含曲线的曲线（表面）类型之间的关系。在一个复维中，我们要么得出函数必须是平滑的，要么它的傅立叶系数衰减得很慢。在高维中，这种现象被认为与波前集的性质有关。相关工作将涉及柯西-黎曼函数的唯一性问题。重点将放在沿非复切曲线无限阶消失的函数的表征上，以及非平坦性假设的相关性，因为它与cr -映射的规律性有关。此外，工作将完成有关从复杂n维空间到自身的全纯映射的基本问题。典型的是雅可比矩阵是具有有限体积像（它们存在）和自同构的统一映射，由于未知的原因，它们不能区分可数密集集，但对离散可数集的作用不同。本研究将应用于对复空间几何的一般理解，以及流形上定义的复偏微分算子。

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.