Diophantine approximation, Nevanlinna theory, and integral points and holomorphic curves in higher-dimensional varieties

PI将研究与高维变元算法相关的几个主题，包括全纯曲线的值分布理论中的相关问题。在一个方向上，他将研究证明具有多个分量的仿射簇上的整点集的有限性的问题，或者更广泛地说是有界的问题。这里的结果将是证明PI关于这类簇上的积分点的猜想的进展，它可以被视为仿射曲线上关于积分点的Siegel定理的高维版本。另外一个新的方面是PI计划使数论中的有效方法适应这种更高维度的环境。我们将探索这些结果在方程式、模变分和算术动力学问题中的应用。其中将使用的工具包括丢番图近似中的施密特子空间定理，这是数论中的一个基本和深入的工具。事实上，在这些结果的证明中使用的一些技术有望产生Schmidt子空间定理本身的变化和改进，例如设置有界次的代数点。从Vojta等人的工作中发现，丢番图近似中的许多陈述，如果表述得当，与复分析的一个分支--Nevanlinna理论中的陈述有很大的相似之处。通过Vojta开发的两个主题之间的词典，经常可以在一个主题中获取陈述和证据，并在另一个主题中开发类似的陈述和证据。以这种方式，PI期望证明奈万林纳理论中的类似结果，其中施密特定理对应于Cartan第二主要定理，并且定性地证明全纯曲线的结果类似于整点的结果。这里提出的研究围绕着数学中最古老和最基本的问题之一：理解有理数或整数中的多项式方程组的解的集合。拟议的研究将在几个基本背景下提供新的结果和技术，从而大大有助于理解这一难题和基本问题。此外，这项研究还对数学的其他领域产生了影响。最值得注意的是，它有望在复杂分析中产生结果，并丰富我们对复杂分析和数论之间深刻联系的理解。

The PI will study several interrelated topics connected with the arithmetic of higher-dimensional varieties, including related questions in the value distribution theory of holomorphic curves. In one direction, he will study the problem of proving the finiteness, or more generally bounding the dimension, of the set of integral points on affine varieties having many components "at infinity". Results here will constitute progress towards proving the PI's conjectures for integral points on such varieties, which may be viewed as higher-dimensional versions of Siegel's theorem for integral points on affine curves. An additional novel aspect is the PI's plan to adapt effective methods in number theory to this higher-dimensional setting. Applications of these results to families of equations, modular varieties, and problems in arithmetic dynamics will be explored. Among the tools used will be the Schmidt subspace theorem from Diophantine approximation, a fundamental and deep tool in number theory. In fact, some of the techniques employed in the proofs of these results are expected to yield variations and improvements of the Schmidt subspace theorem itself, for instance to the setting of algebraic points of bounded degree. From the work of Vojta and others, it has been discovered that many statements in Diophantine approximation, when stated appropriately, bear a strong resemblance to statements in Nevanlinna theory, a branch of complex analysis. Through the dictionary between the two subjects developed by Vojta, it is frequently possible to take statements and proofs in one subject and develop analogous statements and proofs in the other subject. In this manner, the PI expects to prove analogous results in Nevanlinna theory, where Schmidt's theorem corresponds to Cartan's Second Main Theorem, and qualitatively, to prove results for holomorphic curves analogous to results for integral points.The research proposed here revolves around one of the oldest and most fundamental problems in mathematics: understanding the set of solutions to a system of polynomial equations in rational numbers or integers. The proposed research would contribute substantially to understanding this difficult and basic problem by providing new results and techniques in several basic contexts. Moreover, the research has ramifications across other areas of mathematics. Most notably, it is expected to yield results in complex analysis, and to enrich our understanding of the deep links between complex analysis and number theory.