Mathematical Sciences: Geometric Function Theory in Several Complex Variables

该项目所涵盖的数学研究的一般领域涉及几个复杂变量和偏微分方程的理论。一些工作将涉及真正的变量微局部技术，其他部分将集中在不同维度的几个复杂变量空间中的域之间的映射。一个特别值得关注的问题是柯西-黎曼流形的唯一延拓问题。实际上，这些研究涉及到的问题是，一个多变量全纯函数是否可以沿着一个域的各个子流形，比如沿着曲线，是常数，而不是到处都是常数。本工作的第二个重点将集中于在可能不同维数的空间中域之间的全纯映射下保存的集合的性质。很多直观的东西，在一个维度上并不令人惊讶，在更高维度上却被证明是错误的。雅可比矩阵处处为行列式为1的映射可以将无界域映射到有界域上。有些序列在复空间的自同构下是等价的，有些则不是。由于难以想象事态的真实情况，这一领域的进展缓慢。分析在区域上定义的一阶微分算子的基础是确定非齐次d-bar方程何时可以解为闭一形式。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。如果可以证明这一点，它将为在复杂空间中建立微分算子和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.