Nevanlinna Theory, Diophantine Approximation, and Related Topics

所提出的研究是多复变、数论和微分几何的接口，特别是涉及Nevanlinna理论和丢番图几何理论之间的某种神秘关系。本课程将探讨以下主题：丢番图逼近与Nevanlinna理论中的移动目标问题、有界度代数点的丢番图逼近，以及复双曲几何理论中的小林猜想。 该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。数论和复分析是两门古老而重要的学科，在工程、生物、计算机科学等领域都有重要的应用。从历史上看，这两个学科是独立发展的。 最近人们发现它们之间有某种联系，特别是复分析中的Nevanlinna理论和数论中的丢番图近似具有惊人的相似性和联系。 这种相互作用导致了数论和复分析中几个重要的新结果。 希望新发现的关系将彻底改变研究，并将导致这两个主题的重大新进展。 这些进步将对整个数学领域产生巨大影响。

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.