Mathematical Sciences: Geometric Function Theory in Several Complex Variables

这个项目所涵盖的数学研究的一般领域涉及几个复变量和偏微分方程式的理论。一些工作将涉及实变量微局部技术，其他部分将集中在不同维度的几个复变量空间中的区域之间的映射。一个特别值得关注的问题是柯西-黎曼流形上的唯一延拓问题。有效地，这些研究涉及多个变量的全纯函数是否可以沿着区域的各个子流形恒定，例如沿着曲线，而不是处处恒定。这项工作的第二个重点将集中在可能不同维度的空间中的区域之间的全纯映射下保持的集的性质。在一个维度上，许多直观的东西并不令人惊讶，但在更高的维度上，却被证明是错误的。雅可比处处有行列式1的映射可以将无界区域映射到有界域。一些序列在复空间的自同构下是等价的，而另一些则不是。很难看到事态的真实情况，这使得这一领域的进展缓慢。分析定义在区域上的一阶微分算子的基本问题是确定非齐次d-bar方程何时可解为封闭的一种形式。有相当多的证据表明，答案在于互补集的多项式凸性的几何概念。如果能够证明这一点，它将为在复空间中建立微分算子和区域形状之间的深层联系提供一个主要工具。

The general area of mathematical research encompassed by this project relates to the theory of several complex variables and partial differential equations. Some of the work will involve real variable microlocal techniques, other parts will focus on mappings between domains in spaces of several complex variables of different dimensions. One question of particular concern is that of unique continuation in Cauchy-Riemann manifolds. Effectively these studies involve questions of whether or not a holomorphic function of several variables can be constant along various submanifolds of a domain, such as along a curve, and not be constant everywhere. A second thrust of this work will focus on properties of sets preserved under holomorphic mappings between domains in spaces of possibly different dimensions. Much of what is intuitive, and not surprising in one dimension, turns out to be false in higher. Mappings whose Jacobians have determinant one everywhere may map unbounded domains onto bounded ones. Some sequences are equivalent under automorphisms of complex space, others are not. Difficulty in visualizing the true state of affairs makes progress slow in this area. Fundamental to the analysis of first-order differential operators defined on domains is the determination of when the inhomogeneous d-bar equation can be solved for closed one-forms. There is considerable evidence to suggest that the answer lies in the geometric concept of polynomial convexity of the complementary set. If this can be shown to be the case, it will provide a major tool in establishing deep connections between the differential operators and the shape of domains in complex space.