Polynomial Inequalities and Applications

这项研究项目涉及数学分析，特别是在逼近理论、正交多项式和位势理论方面。这些都是数学的一部分，起源于一个多世纪前，为从磁共振成像扫描仪到飞机设计的广泛应用提供了基础和工具。这项工作的一个主要焦点是通过相对透明和可计算的量来理解复杂的数学对象。该项目是这一经典领域的延续，但对其中的一些问题有了新的看法，旨在开发新的方法来解决一些众所周知的开放问题。虽然这是基础研究，但该项目的结果预计将在数学、物理和工程的其他领域有用。预计该项目将激发本科生的兴趣，并为他们改善研究环境。考虑的主要研究领域有四个。第一类是当传统的共形延拓不存在时，共形映射/格林函数的近似延拓。当假设光滑性不如解析性时，可以用这些方法来代替保角延拓。第二个领域涉及新类型的多项式不等式，以解决由光滑的Jordan曲线和圆弧组成的集合上的Bernstein和Markov型不等式中的最佳常数问题，从而结束了一个多世纪以来逼近理论的一个章节。第三个是研究与一般正交多项式族相关的Christoffel-Darboux核，它可能应用于随机矩阵理论中的普适性结果。第四个研究领域是对Widom关于切比雪夫多项式范数的猜想的澄清和解决。各种多项式不等式将这些领域联系在一起，并有望在所研究问题的解决中发挥决定性作用。

This research project concerns mathematical analysis, in particular in approximation theory, orthogonal polynomials, and potential theory. These are parts of mathematics that originated well over a century ago and have provided the foundations and the tools for widespread applications, from magnetic resonance imaging scanners to airplane designs. One main focus of the work is to understand complicated mathematical objects in terms of relatively transparent and computable quantities. The project is a continuation of this classical area, but with a new and fresh look at some of its questions, aimed at developing new methods to solve some well-known open problems. Though it is basic research, the results of the project are anticipated to be useful in other areas of mathematics, physics, and engineering. It is expected that the project will stimulate interest in undergraduates and enhance the research environment for them. There are four main research areas that are considered. The first is the approximative extensions of conformal maps/Green's functions when conventional conformal extensions do not exist. These can be used in lieu of conformal extensions when less smoothness than analyticity is assumed. The second area concerns new types of polynomial inequalities to settle the best constant problem in Bernstein- and Markov-type inequalities on sets consisting of smooth Jordan curves and arcs, thereby closing a chapter in approximation theory that is more than a century old. The third is the study of Christoffel-Darboux kernels associated with general families of orthogonal polynomials with a possible application to universality results in random matrix theory. The fourth research area concerns the clarification and resolution of Widom's conjecture on the norm of Chebyshev polynomials. Various polynomial inequalities connect these fields and are expected to play a decisive role in the solutions of the problems under study.