Nevanlinna Theory, Diophantine Approximation, and Related Topics

所提出的研究存在于多个复变量、数论和微分几何的界面上，特别是涉及到纳瓦林纳理论和丢番图几何理论之间的某种神秘的关系。我们将研究以下问题：丢番图逼近中的运动目标问题、有界阶代数点的丢番图逼近以及复双曲几何理论中的Kobayashi猜想。项目的第二部分将涉及研究n维欧氏空间中极小曲面和极小子流形的高斯映射的值分布性质。数论和复数分析是两门古老而又重要的学科，在工程、生物学、计算机科学等领域有着重要的应用。从历史上看，这两门学科是独立发展的。近年来，人们发现它们之间存在着某种程度上的联系，特别是复变分析中的nevanlinna理论和数论中的丢番图近似有着惊人的相似和联系。这种相互作用在数论和复数分析中产生了几个重要的新结果。人们希望这一新发现的关系将给研究带来革命性的变化，并将在这两个学科中引领重大的新进展。这样的进步将对整个数学领域产生重大影响。

The proposed research lies in the interface of several complex variables, number theory, and differential geometry, in particular, it involves the somewhat mysterious relationship between Nevanlinna theory and the theory of Diophantine geometry. The following topics will be investigated: moving target problems in Diophantine approximation and Nevanlinna theory, Diophantine approximation by algebraic points of bounded degree, and the Kobayashi conjecture in the theory of complex hyperbolic geometry. A second part of the project will involve the study of the value distribution properties of the Gauss map of minimal surfaces and minimal submanifolds in the n-dimensional Euclidean space. Number theory and complex analysis, two old but important subjects, have been found to have important applications in Engineering, Biology, Computer Sciences, and other fields. Historically these two subjects developed independently. Recently it was discovered that they are somewhat related; in particular, Nevanlinna theory in complex analysis and Diophantine approximation in number theory bear striking similarities and connections. This interplay has lead to several important new results in number theory and in complex analysis. It is hoped that the newly discovered relationship will revolutionize the researches and will lead major new advances in both subjects. Such advances would have a great impact in the whole area of mathematics.