Hermitian Analysis and CR Geometry

PI将继续学习分析和几何，因为它与复数有关。复数和复变函数在纯数学的其他研究领域和数学应用中已成为非常有用的工具。例如，研究机翼上方的气流使用复数和复数函数来模拟这种流动。在数学科学中，这个主题仍然是一个至关重要的话题。意识到这一点，PI最近开始教授所谓的高等工程数学。讲授这份材料精确地展示了拟议的研究如何开发出目前在物理和工程学中使用的基础数学的更高维度的类比。除了数学研究，国际工程学院还将与伊利诺伊大学厄巴纳-香槟分校的工程学院合作，微调高等工程数学的课程。埃尔米特分析起源于19世纪的数学。这门学科始于两个世纪前傅里叶关于热扩散的工作。它是利用希尔伯特空间理论发展起来的，通过它在量子力学中的应用，以及它在信号处理中的作用。现代厄米特分析依赖于复分析和CR几何，并为其提供信息，这是PI的主要研究领域。现代观点强调的工具是正交化展开和正交投影。PI引入了一种称为正交齐化的代数技术，它将这些想法与球之间适当映射的几何联系起来。PI将他关于球的适当图像的体积的较早结果之一重新表述为变分问题，从而将结果扩展到更一般的情况。PI关于不同维复空间中球之间的适当映射的工作也揭示了表示理论和代数组合学之间意想不到的联系。本研究的主要目的是将正交齐化的思想推广到有理情形，建立附加的变分不等式，研究余维为正的真全纯映射的同伦，并进一步发展CR复杂性这一主题。在这一过程中，国际数学家协会将继续就这些主题指导青年数学家，并组织和出席会议。

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.