Geometric Theory of Meromorphic Functions

作者打算继续研究亚纯函数理论中的几何问题，使用他以前工作中开发的新技术。主要研究方向如下：a)与Bloch定理和单连通黎曼曲面的类型问题有关的问题，特别是曲面的共形型与其积分曲率之间的关系；b)实代数几何中的几何函数论问题。更具体地说，它包括计算某些几何起源的代数方程组的实数解，这些方程在线性控制理论中有重要的应用。C)将几何函数论的结果推广到任意维空间中的拟正则映射。d)射影空间中全纯曲线族的正规性准则。数学及其应用中的一个基本问题是给定的方程或方程组是否有解，有多少解，解的位置在哪里。在亚纯函数理论中，有人研究了方程f(Z)=a的这类问题，其中a是给定的复数，f是给定的亚纯函数。亚纯函数类包括初等函数，如有理函数、指数函数、三角函数等，也包括特殊函数，又称高阶超越函数，如Gamma函数、Ary函数、椭圆函数等。大多数数学应用中的函数都属于这类函数。在现代数学中，关于方程可解性的问题通常是用几何语言来表示的，这使得结果更符合我们的几何直觉。数学及其应用的发展逻辑要求将结果推广到称为“全纯曲线”的矢量值函数。作者计划继续他对亚纯函数和全纯曲线的几何理论的研究，其中一部分涉及到实解的存在，这比通常研究的复解的存在要微妙得多。这一部分的灵感来自于所谓的“极点配置问题”，它是线性系统控制理论中一个尚未解决的重大数学问题。这方面的结果将对复杂的自动控制系统的设计具有一定的指导意义。这些结果将对通过特定类别的控制设备控制给定大小的系统的可能性建立限制。

The proposer intends to continue his study of geometric questions in the theory of meromorphic functions, using the new techniques developed in his previous work. The main directions of the proposed research are the following.a) Questions related to Bloch's theorem and the Type Problem ofa simply connected Riemann surface, especially the relationsbetween the conformal type of a surface and its integral curvature.b) Problems of geometric function theory arising in real algebraicgeometry. More specifically, it includes counting real solutionsof certain systems of algebraic equations of geometric origin, whichhave important applications in linear control theory. c) Generalization of results of geometric function theory toquasiregular maps in spaces of arbitrary dimension.d) Normality criteria for families of holomorphic curves in projectivespaces.One of the basic questions in mathematics and its applications iswhether a given equation or a system of equations has solutions,how many, and where are they located. In the theory of meromorphic functionsone studies these questions for equations of the typef(z)=a, where a is a given complex number and f a given meromorphic function.The class of meromorphic functions includes elementary functions,such as rational, exponential and trigonometric ones, as well as the specialfunctions, a. k. a. higher transcendental functions, such as theGamma function, Airy functions, elliptic functions and so on.Most functions arising in applications of mathematics belong tothis class. In modern mathematics, questions about solvabilityof equations are usually formulated in geometric language, which makesthe results appealing to our geometric intuition. The logic of development of mathematics and its applicationsrequire an extension of results to vector-valued functions knownas ``holomorphic curves''. The proposer plans to continue his studyof geometric theory of meromorphic functions and holomorphic curves.A part of the proposal is related to existence of real solutions, which isby far more subtle than the existence of complex solutions, which are usuallystudied. This part is inspired by the so-called "pole placement problem", which is a major unsolved mathematicalproblem in control theory of linear systems. The results in this areawill have implications for the design of complicated automatic control systems. These results would establish limitations on the possibility tocontrol a system of given size by a control device of certain class.