Problems in geometric function theory

几何函数论问题

• 批准号: 1361836
• 负责人: Alexandre Eremenko
• 金额: $ 17.29万
• 依托单位: Purdue University
• 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: 2014-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361836&HistoricalAwards=false
• 关键词: Problems; geometric; function; theory

Analytic functions are objects with very nice properties that allow one to use a multitude of techniques to explore them, and they constitute the most important class of functions used in mathematics and its applications to science and engineering. The choice of problems of the theory of analytic functions in this proposal is motivated by their intrinsic mathematical interest, rather than specific applications. The problems chosen for this proposal are old and difficult, and the PI proposes to study them using a variety of new methods recently developed by him and his collaborators, as well as by the other research groups. Most of these problems arise in the previous research of the PI funded by NSF. This previous research led to several significant results relevant to applications of mathematics in control theory, physics and material science. They also had impact on education: advanced mathematical courses on graduate level, PhD theses supervised by the PI and training of postdoctoral scholars. The present proposal is expected to have a similar impact.The main topic of this proposal is the geometric problem of classification of spherical polygons, and more generally, of metrics of constant positive curvature with conical singularities. This is closely related to the study of analytic functions defined by the Heun equation, the linear second order differential equation with four regular singularities. These functions lie on the boundary of the set of Special functions of Mathematical physics, but much less is known about them in comparison with classical special functions which are defined by equations with at most three singularities. The goal of this research is to classify spherical quadrilaterals up to isometry. The methods proposed are those used in the recent research of PI, A. Gabrielov, E. Mukhin, V. Tarasov and A. Varchenko on the Shapiro conjecture in real algebraic geometry.The second subject of proposed research is value distribution of holomorphic curves in projective spaces, especially of Brody curves, and curves defined by higher order linear differential equations. One specific goal is to prove a recent conjecture of Duval and da Costa on the Second Main theorem for Brody curves. The PI plans to use potential-theoretic methods developed in his previous work. The third topic is investigation of a class of extremal problems of new type for holomorphic functions in the unit disc that are inspired by questions in Control theory, namely the question of stabilizability of several systems by single output feedback control device. The proposed methods are those employed in the previous research on this problem by PI, jointly with W. Bergweiler.

解析函数是具有非常好的属性的对象，允许人们使用多种技术来探索它们，并且它们构成了数学及其在科学和工程中的应用中使用的最重要的一类函数。本提案中解析函数理论问题的选择是出于其内在的数学兴趣，而不是具体的应用。该提案选择的问题既古老又困难，PI 建议使用他和他的合作者以及其他研究小组最近开发的各种新方法来研究这些问题。这些问题大多出现在NSF资助的PI前期研究中。先前的研究取得了一些与数学在控制理论、物理学和材料科学中的应用相关的重要成果。它们还对教育产生了影响：研究生阶段的高级数学课程、由 PI 指导的博士论文以及博士后学者的培训。目前的提案预计将产生类似的影响。该提案的主要主题是球面多边形分类的几何问题，更一般地说，是具有圆锥奇点的恒定正曲率度量的几何问题。这与 Heun 方程（具有四个正则奇点的线性二阶微分方程）定义的解析函数的研究密切相关。这些函数位于数学物理特殊函数集的边界上，但与由最多三个奇点的方程定义的经典特殊函数相比，人们对它们的了解要少得多。本研究的目标是将球面四边形分类为等距。所提出的方法是PI、A. Gabrielov、E. Mukhin、V. Tarasov和A. Varchenko最近研究实代数几何中的夏皮罗猜想所使用的方法。所提出的研究的第二个主题是射影空间中全纯曲线的值分布，特别是布罗迪曲线和高阶线性微分方程定义的曲线。一个具体目标是证明杜瓦尔和达科斯塔最近关于布罗迪曲线第二大定理的猜想。 PI 计划使用他之前工作中开发的势理论方法。第三个课题是受到控制理论问题的启发，研究一类新型单位圆盘中全纯函数的极值问题，即单输出反馈控制装置的多个系统的稳定性问题。所提出的方法是 PI 与 W. Bergweiler 合作之前针对该问题的研究中所采用的方法。