Meromorphic Functions and Holomorphic Curves

亚纯函数和全纯曲线

• 批准号: 9800084
• 负责人: Alexandre Eremenko
• 金额: $ 8.86万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800084&HistoricalAwards=false
• 关键词: Meromorphic; Functions; Holomorphic; Curves

Proposal: DMS-9800084 Principal Investigator: Alexandre Eremenko Abstract: A. Eremenko will continue his study of the distribution of values of meromorphic functions and their generalization, holomorphic maps from the complex line to projective spaces, using the new methods of potential theory developed in his previous work. With this approach he already obtained such results as the proof of a defect relation for non-linear hypersurfaces conjectured by B. Shiffman (joint work with Sodin, 1992) and the proof of the Modified Cartan conjecture in dimension three (1996). Specifically, he plans to concentrate on the following topics: normality criteria for holomorphic maps omitting hyperplanes, the study of extremal maps for defect relations, and the Inverse Problem of the Value Distribution Theory for maps to projective spaces, with special attention to the maps satisfying algebraic differential equations, both linear and non-linear. Meromorphic functions constitute the most basic class of functions used in mathematics and its applications. This class includes elementary functions like the exponential, cosine and tangent functions, as well as higher transcendental functions, like the Gamma function, Airy functions, elliptic functions, etc., which are indispensable in physics, engineering and other applications. The fundamental problem about these functions is the study of solutions of equations f(x)=a, where f is a given function, and a is a given complex number. Usually such equations have infinitely many solutions and the question is how their location depends on the right-hand side of the equation. This is the subject of Value Distribution Theory, a great achievement of mathematical analysis in the second half of the nineteenth century and the first half of the present century. The logic of development of mathematics and potential applications to differential equations require an extension of the main results of this theory to the case of functions whose values are no longer numbers, but i nstead are vectors. Despite major efforts by several outstanding researchers in the beginning of the twentieth century, our current understanding of this subject is very incomplete. Recent breakthroughs give hope of major new advances in the area. Such advances would extend our knowledge of a basic and fundamental mathematical subject, which is the value distribution of the vector-valued entire functions known as "holomorphic curves in projective spaces."

摘要：A. Eremenko将继续他的亚纯函数的值分布及其推广，从复线到射影空间的全纯映射的研究，使用他之前工作中发展的势理论的新方法。通过这种方法，他已经获得了诸如证明B. Shiffman猜想的非线性超曲面的缺陷关系（与Sodin合作，1992年）和证明三维的修正Cartan猜想（1996年）等结果。具体来说，他计划集中在以下主题：省略超平面的全纯映射的正规准则，缺陷关系的极值映射的研究，以及射影空间映射的值分布理论的反问题，特别关注满足线性和非线性代数微分方程的映射。亚纯函数是数学及其应用中最基本的一类函数。本课程包括指数函数、余弦函数、正切函数等初等函数，以及伽玛函数、艾里函数、椭圆函数等在物理、工程等应用中不可或缺的高等超越函数。这些函数的基本问题是研究方程f(x)=a的解，其中f是一个给定的函数，a是一个给定的复数。通常这样的方程有无穷多个解，问题是它们的位置如何依赖于方程的右边。这就是价值分配理论的主题，价值分配理论是19世纪下半叶和本世纪上半叶数学分析的伟大成就。数学发展的逻辑和微分方程的潜在应用需要将这一理论的主要结果扩展到函数的情况，这些函数的值不再是数字，而是向量。尽管几位杰出的研究人员在20世纪初做出了重大努力，但我们目前对这一主题的理解非常不完整。最近的突破给这一领域带来了重大新进展的希望。这样的进步将扩展我们对一个基本和基本的数学主题的知识，这就是被称为“射影空间中的全纯曲线”的向量值整个函数的值分布。