New Frontiers in Several Complex Variables

微积分理论使人们能够精确地理解连续变化的大小，这构成了许多现代科学和技术的数学基础。复数是一种广义数系，为代数运算提供了更大的灵活性，例如允许提取负量的平方根。在这个项目中，研究人员将应用微积分的工具来平滑地改变依赖于一个或多个复杂输入的复杂输出。这样的对象在数学文献中称为全纯函数。全纯函数的研究在各种现象中都有应用，如液体流动、热传导、素数分布和计算机程序的速度。该项目将使用两种方法来构造全纯函数。第一种是通过一个称为非齐次柯西-黎曼方程的方程组。第二种是投影法，它将全纯函数的集合几何地解释为一个更大的函数集合中的一个子集合，并利用这种几何洞察力来构造具有特定性质的全纯函数。在这个项目中对全纯函数的更深入的理解将对数学内部和外部的应用都是有用的。该项目还将包括本科生参与者，他们将研究问题的计算和理论方面的工作。这样的参与将使学生接触到具有挑战性的数学问题，并将有助于培养未来的数学家和科学家。一个或多个复变量的全纯函数构成分析中一类自然而重要的函数，在科学和工程中具有重要的应用。这个项目研究了构造这类函数的两种经典技术：非齐次柯西-黎曼方程和投影算子。对于Cauchy-Riemann方程，主要研究这些方程在环空中的解，其中复欧氏空间中的环空是通过去掉紧致子集(空洞)而从更大的区域(包络)中获得的区域。由于环不是全纯域，在这种情况下出现了一些新的现象，这些现象在人们熟知的伪凸域和经典的伪凸域中是不存在的。对环的复杂分析和亚历山大对偶(拓扑学中的一个思想圈)之间的类比有望形成对这些问题的新的攻击方法。与基于偏微分算子先验估计的经典方法相结合，这些新方法将被用于产生关于新的非伪凸域的函数论的重要新信息。关于通过投影算子的几何构造，主要的研究焦点将是关于非光滑区域上的勒贝格空间中的Bergman投影及其相关的投影算子的正则性的一类问题。这种规律性以一种尚未完全理解的方式依赖于域的几何形状。在要研究的领域中，有有限群的球的商，以及边界包含原点的莱因哈特域。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The theory of calculus, which makes possible a precise understanding of continuously varying magnitudes, forms the mathematical underpinning of much of modern-day science and technology. The complex numbers are a system of generalized numbers providing greater flexibility with algebraic operations, such as allowing for the extraction of square roots of negative quantities. In this project, the investigator will apply the tools of calculus to smoothly varying complex outputs which depend on one or more complex inputs. Such objects are known in the mathematical literature as holomorphic functions. The study of holomorphic functions has applications to phenomena as diverse as the flow of liquids, heat conduction, the distribution of prime numbers, and the speed of computer programs. The project will employ two methods to construct holomorphic functions. The first is via a system of equations called the inhomogeneous Cauchy-Riemann equations. The second, the method of projections, interprets the collection of holomorphic functions geometrically as a sub-collection inside a larger collection of functions, and uses this geometric insight to construct holomorphic functions with specified properties. The deeper understanding of holomorphic functions to be gained in this project will be useful for applications both within and outside mathematics. The project will also involve undergraduate participants, who will work on both the computational and theoretical aspects of the problems. Such participation will expose students to challenging mathematical problems and will contribute to the training of future mathematicians and scientists.Holomorphic functions of one or several complex variables constitute a natural and important class of functions in analysis and have important applications throughout science and engineering. This project investigates two classical techniques for constructing such functions: the inhomogeneous Cauchy-Riemann equations and projection operators. For the Cauchy-Riemann equations, the main thrust is to study the solution of these equations in annuli, where an annulus in complex Euclidean space is a domain obtained from a larger domain (the envelope) by removing a compact subset (the hole). Since annuli are not domains of holomorphy, new phenomena appear in this situation which are not found in the well-understood and classical setting of pseudoconvex domains. The analogy between complex analysis on annuli and Alexander duality (a circle of ideas in topology) is expected to forge novel methods of attack on these questions. Coupled with classical approaches based on a priori estimates for partial differential operators, these new methods will be employed to yield important new information on function theory of new classes of non-pseudoconvex domains. With regards to the geometric construction via projection operators, a primary focus of investigation will be a class of questions concerning the regularity of the Bergman projection, and other associated projection operators, in Lebesgue spaces on non-smooth domains. Such regularity depends on the geometry of the domain in ways that are not completely understood. Among the domains to be studied are quotients of balls by finite groups, and Reinhardt domains whose boundary contains the origin.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.