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一类函数。在这项建议中，选择解析函数论的问题是出于其内在的数学兴趣，而不是具体的应用。为这项建议选择的问题既古老又困难，国际和平协会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以往国家自然科学基金资助的PI研究中。这项先前的研究导致了与数学在控制理论、物理和材料科学中的应用相关的几个重要结果。它们还对教育产生了影响：研究生水平的高级数学课程、由PI指导的博士论文以及博士后学者的培训。这一建议的主要主题是球面多边形的分类几何问题，更广泛地说，是具有锥形奇点的常正曲率度量的几何问题。这与Heun方程所定义的解析函数的研究密切相关，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。这项研究的目标是将球面四边形分类到等距。所提出的方法是PI，A.Gabrielov，E.Mukhin，V.Tarasov和A.Varchenko最近对实代数几何中Shapiro猜想的研究中使用的方法。第二个研究主题是射影空间中全纯曲线的值分布，特别是Brody曲线，以及由高阶线性微分方程定义的曲线的值分布。一个具体的目标是证明Duval和da Costa最近关于Brody曲线的第二主要定理的一个猜想。PI计划使用他之前工作中开发的势能理论方法。第三个主题是研究单位圆盘上一类新的全纯函数极值问题，这类极值问题是由控制论中的问题引起的，即多个系统的单输出反馈控制器镇定问题。所提出的方法是PI与W·伯格韦勒在以前对这个问题的研究中所采用的方法。

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.