Hermitian Analysis and CR Geometry

PI将继续他的分析和几何研究，因为它与复数有关。复数和复变量函数已经成为研究纯数学其他领域和数学应用的非常有用的工具。例如，研究机翼上的气流使用复数和复数函数来模拟气流。这一课题在数学科学中仍然是一个至关重要的课题。意识到这一点，PI最近开始教授所谓的高等工程数学。教学这些材料在一个精确的意义上显示了所提议的研究如何发展高维的基础数学，目前在整个物理和工程中使用。除了数学研究，PI还将与伊利诺伊大学厄巴纳-香槟分校的工程学院合作，微调高等工程数学课程。厄米分析起源于19世纪的数学。两个世纪前，傅里叶对热扩散的研究开始了这一课题。它利用希尔伯特空间理论，通过它在量子力学中的应用，以及它在信号处理中的作用而发展起来。现代厄米分析既依赖于并通知复杂分析和CR几何，PI的主要研究领域。现代的观点强调标准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，将这些思想与球之间的固有映射的几何联系起来。PI重新表述了他的一个老的结果，关于球的适当图像的体积，作为一个变分问题，从而将结果扩展到更一般的情况。PI对不同维度复杂空间中球之间的适当映射的研究也揭示了与表示理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，研究正余维纯纯映射的同伦问题，并进一步发展CR复杂度问题。在此过程中，PI将继续在这些主题上指导年轻数学家，并组织和参加会议。

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.