Mathematical Sciences: Diophantine Approximations of Algebraic Points

本课题研究全纯函数的丢番图逼近和值分布理论。P·沃伊塔、C·奥斯古德等人的研究表明，这两个领域实际上通过形式上的类比而联系在一起。这种观察导致了新的猜想，旧定理的较短证明，以及新定理的证明。该项目将继续这一趋势，主要在丢番图近似领域工作。证明可以简化的结果包括关于有理点在有限约化的域上的有理点的有限定理和关于主极化的阿贝尔簇的有限的Faltings定理。可以证明的新结果包括“运动目标”类型的结果，即涉及缓慢变化的一族方程的丢番图近似陈述。一个遥远的目标是“ABC猜想”。使用图厄方法的校样的技术改进也将继续进行。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

This project deals with diophantine approximation and value distribution theory of holomorphic functions. Work by P. Vojta, C. Osgood, and others has shown that these two fields are in fact connected by a formal analogy. This observation has led to new conjectures, shorter proofs of old theorems, and proofs of new theorems. This project will continue this trend, primarily working in the field of diophantine approximation. Results whose proofs may be simplified include finiteness theorems for rational points on abelian varieties and Faltings' theorem on the finiteness of principally polarized abelian varieties over a given field with finite reduction. New results which may be proved include "moving targets" type results; i.e., diophantine approximation statements which involve a slowly varying family of equations. A distant goal is the "abc conjecture." Technical improvements in proofs using Thue's method will also be pursued. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.