Topics in extremal problems in complex analysis and potential theory

该项目的目标是为复杂分析和势理论中的深层极值问题提供新的理论见解。特别强调的是对称方法和二次微分方法，这是PI在他以前的作品中发展出来的新版本。PI将结合丰富多样的现有工具，对一些具有挑战性的老问题以及最近出版物中提出的一些有趣的新问题进行测试。该项目的三个主题是：(i)发展朗道型的覆盖定理和分析函数的最小面积问题；（ii）获得平面构型共形不变量（如容量、谐波测度和双曲密度）的新的尖锐估计，并可能应用于共形映射的边界行为问题，如著名的布伦南猜想；（iii）在Sendov猜想、Smale猜想等著名的复数多项式问题中，寻找一种建立多项式极值零点对称性或临界点的方法。PI预计这项研究将涉及新方法的发展，这反过来将增强我们对复杂分析和势理论中极值问题理论的理解。所提出的方法旨在开发复杂分析的创新工具，这些工具将至少在两个方面潜在地应用于科学和工程的其他特定领域。首先，二次微分理论的结果影响了数学的其他分支，并在理论物理中有应用，特别是弦理论。其次，准确估计平面结构的功能特征，如半平面容量、扭转刚度和主频率，在共形不变过程理论和数学物理中是重要的。PI正在与一个不断扩大的核心小组合作，这个小组由敬业和成熟的研究生组成。他期望这个项目，特别是关于复数多项式的问题，虽然是初级的，但非常深刻，为他们提供有趣的研究课题。由于德州理工大学在地理位置上位于德克萨斯州西部的农村地区，人口服务不足，代表性不足，因此该项目与研究生合作的机会之一将是吸引更多来自周围人口的多样化和更合格的研究生从事数学科学的职业，并使他们在毕业后更容易获得良好的职位。

The goal of the project is to provide new theoretical insight regarding deep extremal problems in complex analysis and potential theory. Particular emphasis will be placed on the method of symmetrization and the method of quadratic differentials, new versions of which were developed by the PI in his previous works. The PI will test these innovative techniques, in combination with a rich variety of existing tools, on some old challenging problems as well as on some new interesting questions posed in recent publications. The three main themes of the project are: (i) Developing covering theorems of Landau type and minimal area problems for analytic functions; (ii) Obtaining new sharp estimates for conformal invariants (such as capacities, harmonic measure, and hyperbolic density) of planar configurations with possible applications to problems on the boundary behavior of conformal mappings such as a well known Brennan's conjecture; (iii) Searching for a method to establish symmetry of zeros or critical points of extremal polynomials in some well known problems on complex polynomials such as Sendov's conjecture and Smale's conjecture. The PI anticipates that this study will involve the development of new approaches, which in turn will enhance our understanding of the theory of extremal problems in complex analysis and potential theory.The proposed methodology is targeted towards developing innovative tools in complex analysis which will have potential applications to other specific areas in science and engineering on at least two fronts. First, results in the theory of quadratic differentials impacts other branches of mathematics and has applications in theoretical physics, in particular, String Theory. Second, accurate estimates of functional characteristics of planar configurations, such as half-plane capacity, torsional rigidity, and principal frequency, are important in the theory of conformally invariant processes and in mathematical physics. The PI is working with an expanding core group of engaged and maturing graduate students. He expects the project, especially the problems about complex polynomials, which although elementary to state are very deep, to provide interesting research topics for them. Because Texas Tech University is geographically situated in rural West Texas with its under-served and under-represented populations, one of the opportunities for this project with its engagement of graduate students will be to attract more diverse and better qualified graduate students to careers in the mathematical sciences from the surrounding populations and ease placing them in good positions upon graduation.