Problems in geometric function theory

解析函数是具有非常好的性质的对象，允许人们使用多种技术来探索它们，它们构成了数学及其在科学和工程中的应用中最重要的函数类。在本提案中，解析函数理论问题的选择是由其内在的数学兴趣驱动的，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它们。这些问题大多出现在NSF资助的PI的前期研究中。这项先前的研究导致了几个与数学在控制理论、物理和材料科学中的应用相关的重要结果。他们也对教育产生了影响：研究生水平的高等数学课程，PI指导的博士论文和博士后学者的培训。预计目前的建议也会产生类似的影响。本提案的主要主题是球面多边形的分类几何问题，更一般地说，是具有圆锥奇点的常正曲率度量的分类几何问题。这与研究由Heun方程定义的解析函数密切相关，Heun方程是具有四个正则奇点的线性二阶微分方程。这些函数位于数学物理的特殊函数集的边界上，但与由至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。本研究的目的是对球面四边形进行等距分类。所提出的方法是PI、A. Gabrielov、E. Mukhin、V. Tarasov和A. Varchenko在实际代数几何中的夏皮罗猜想的最新研究中使用的方法。本文研究的第二个主题是全纯曲线在射影空间中的值分布，特别是Brody曲线和由高阶线性微分方程定义的曲线。一个具体的目标是证明Duval和da Costa最近关于Brody曲线第二主要定理的猜想。PI计划使用在他以前的工作中开发的势理论方法。第三个课题是研究受控制论问题，即单输出反馈控制装置对多个系统的稳定性问题启发的一类单位圆盘上全纯函数的新型极值问题。所提出的方法是PI与W. Bergweiler在之前的研究中所使用的方法。

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.