Analysis and Dynamics in Several Complex Variables

复数是真实的数的扩展，在科学、工程和经济学中不可或缺。例如，由交流电驱动的电路，例如向家庭供电的发电机中的电路，可以使用复数进行分析。这个项目是在复数和复值函数的设置。微积分的基本定理是以真实的数为背景的，它使用积分公式来建立函数和它们的导数之间的联系。 主要研究者将研究在高维空间或更一般的空间中使用积分公式表示的复值函数的光滑性，并考虑该理论的一些应用。该项目还将通过论文工作支持对研究生的培训。本项目有四个研究课题。主要研究者将研究严格伪凸域上的柯西-黎曼方程的正则性，或者在复杂流形中具有足够数量的正或负Levi特征值的域。该研究将集中在最小的正则性要求的域，同时寻求解决方案，具有高阶正则性。更一般地说，主要研究者将研究复微分形式的积分表示，并使用它来研究严格伪凸或Z（1）域上复杂结构的变形稳定性。主要研究者将使用复分析和动力系统的方法研究紧致复流形的邻域分类，并将继续研究复欧几里得空间中真实的子流形的CR奇异性。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Complex numbers, an extension of real numbers, are indispensable in science, engineering, and economics. For instance, a circuit driven by an alternating current, such as that in a generator that delivers power to a household, can be analyzed using complex numbers. This project is in the setting of complex numbers and complex-valued functions. The fundamental theorem of calculus, which is in the setting of real numbers, uses an integral formula to establish a connection between functions and their derivatives. The principal investigator will study the smoothness of complex-valued functions using an integral formula representation in higher-dimensional spaces, or in more general spaces, and consider some applications of this theory. The project will also support the training of graduate students through their thesis work. This project has four research topics. The principal investigator will study the regularity of Cauchy-Riemann equations on strictly pseudoconvex domains, or domains that have a sufficient number of positive or negative Levi eigenvalues in a complex manifold. The research will focus on minimum regularity requirements for the domains while seeking solutions that have higher-order regularity. More generally, the principal investigator will study the integral representation of complex differential forms and use it to study the stability of deformation of complex structures on strictly pseudoconvex or Z(1) domains. The principal investigator will study the classification of neighborhoods of a compact complex manifold using methods from complex analysis and dynamical systems, and will continue to study the CR singularity of real submanifolds in complex Euclidean spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.