Hermitian Analysis and CR Geometry

埃尔米特分析和 CR 几何

• 批准号: 1361001
• 负责人: John D'Angelo
• 金额: $ 35.16万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361001&HistoricalAwards=false
• 关键词: Hermitian; Analysis; CR; Geometry

The PI will continue his study of analysis and geometry as it relates to complex numbers. Complex numbers and functions of complex variables have become very useful tools in the study of other areas of pure mathematics and in the application of mathematics. For instance, the study of airflow over a wing uses complex numbers and complex functions to model this flow. The subject continues to be a vital and important topic in the mathematical sciences. Realizing this, the PI has recently begun teaching what is called Advanced Engineering Mathematics. Teaching this material shows in a precise sense how the proposed research develops higher dimensional analogues of fundamental mathematics currently used throughout physics and engineering. In addition to the mathematical research, the PI will also work with the Engineering College at the University of Illinois at Urbana-Champaign on fine-tuning the curriculum in Advanced Engineering Mathematics.Hermitian analysis has its roots in 19th century mathematics. The subject began, two centuries ago, with the work of Fourier on heat diffusion.It has developed using the theory of Hilbert spaces, by its applications to quantum mechanics, and via its role in signal processing. Modern Hermitian analysis both relies upon and informs complex analysis and CR geometry, the PI's primary research areas. The modern point of view emphasizes the tools of orthonormal expansion and orthogonal projection. The PI has introduced an algebraic technique called orthogonal homogenization which connects these ideas to the geometry of proper mappings between balls. The PI has reformulated one of his older results, on volumes of proper images of balls, as a variational problem, thereby extending the results to considerably more general situations. The PI's work on proper mappings between balls in different dimensional complex spaces has also uncovered unexpected connections to representation theory and algebraic combinatorics. The primary purposes of this research are to extend the ideas of orthogonal homogenization to the rational case, to establish additional variational inequalities, to study homotopy for proper holomorphic mappings of positive codimension, and to further develop the subject of CR complexity. In the process the PI will continue to mentor young mathematicians in these topics and to organize and attend conferences.

PI将继续他的分析和几何学的研究，因为它涉及到复数。 复数和复变函数在纯数学的其他领域的研究和数学应用中已经成为非常有用的工具。例如，机翼上气流的研究使用复数和复变函数来模拟这种流动。这个问题仍然是一个至关重要的和重要的主题在数学科学。 意识到这一点，PI最近开始教授所谓的高等工程数学。教授这些材料在精确的意义上显示了拟议的研究如何开发目前在物理和工程中使用的基础数学的高维类似物。除了数学研究，PI还将与伊利诺伊大学厄巴纳-香槟分校工程学院合作，微调高等工程数学课程。埃尔米特分析起源于19世纪数学。这门学科始于两个世纪前傅立叶对热扩散的研究，后来发展到希尔伯特空间理论，并通过其在量子力学中的应用，以及在信号处理中的作用。现代埃尔米特分析都依赖于和通知复杂的分析和CR几何，PI的主要研究领域。现代的观点强调正交展开和正交投影的工具。PI引入了一种称为正交均匀化的代数技术，将这些想法与球之间的适当映射的几何学联系起来。PI已经重新制定了他的一个旧的结果，卷适当的图像的球，作为一个变分问题，从而扩大了结果相当更一般的情况。 PI在不同维复杂空间中球之间的适当映射的工作也揭示了与表示论和代数组合学的意想不到的联系。本研究的主要目的是将正交均匀化的思想推广到有理情形，建立额外的变分不等式，研究正余维真全纯映射的同伦，并进一步发展CR复杂性。在此过程中，PI将继续指导这些主题的年轻数学家，并组织和参加会议。