CAREER: Integral Points on Varieties and Related Tools and Topics

了解一组多项式方程的整数解是数学中最古老和最基本的问题之一。这门学科也与计算机科学的主题密切相关。基本问题包括确定积分解集是否是有限的，如果是，如何显式计算这样的解集。强大的几何思想和概念的引入使这门学科取得了重大进展，对基本问题有了更广泛和更深入的理解。从这个几何的角度来看，这个研究项目的目的是通过大量扩展已知的结果、技术和工具来增强我们对这些困难和基本问题的理解。除了研究活动，PI还将为密歇根州立大学精选的一批本科生组织和发起师生研究项目。研究项目将通过本科生系列座谈会和为本科生资助的会议旅行来加强。PI将继续密切参与研究生教育和指导研究生和博士后，建议与拟议的研究链有关的问题。PI将围绕变元上的积分点研究几个相互关联的问题。首先，PI将沿着PI猜想的思路，研究将曲线上的积分点的Siegel定理推广到高维变体的问题。在第二个项目中，PI将研究曲线上的有界次积分点。然后，在不同的方向上，我们将研究有效计算簇上的积分点的问题的各个方面，包括曲线和高维簇。研究簇上的积分点(或有理)点的基本工具来自丢番图逼近。其中一个关键工具就是施密特子空间定理。PI将继续他最近在证明Schmidt定理的推广方面的工作，最终目标是代数点的Schmidt-Wirsing类型猜想。PI将研究算术动力学中的几个“一致有界性”猜想，这是一个非常适合应用上述技术和结果的学科。最后，通过Vojta等人的工作，丢番图近似中的陈述和定理与Nevanlinna理论中的陈述和定理之间存在着惊人的对应关系。国际和平研究所计划研究纳瓦林纳理论中上述问题的类比，以期丰富这两个学科。

Understanding the set of integer solutions to a system of polynomial equations is one of the oldest and most basic problems in mathematics. The subject is also closely related to topics in computer science. Fundamental questions include determining whether or not the set of integral solutions is finite, and if so how to explicitly compute the set of such solutions. The introduction of powerful geometric ideas and concepts into the subject has led to major advances and a wide-ranging and deeper understanding of the basic problems. From this geometric viewpoint, this research project aims to enhance our understanding of these difficult and fundamental problems by substantially extending the known results, techniques, and tools. In addition to research activities, the PI will organize and initiate student-faculty research projects for a select group of undergraduate students at Michigan State University. The research projects will be enhanced by an undergraduate colloquium series and funded conference travel for the undergraduate students. The PI will continue his close involvement with graduate education and mentor graduate students and postdocs, suggesting problems related to the proposed research strands.The PI will study several interrelated problems revolving around the study of integral points on varieties. First, the PI will examine the problem of generalizing Siegel's theorem for integral points on curves to higher-dimensional varieties, along the lines of the PI's conjectures. In a second project, the PI will study integral points of bounded degree on curves. Then, in a different direction, aspects of the problem of effectively computing integral points on varieties will be studied, both for curves and higher-dimensional varieties. Fundamental tools for studying integral (or rational) points on varieties come from the subject of Diophantine approximation. One such key tool is Schmidt's Subspace Theorem. The PI will continue his recent work on proving generalizations of Schmidt's theorem, with a Schmidt-Wirsing type conjecture for algebraic points as an ultimate goal. The PI will study several "uniform boundedness" conjectures in arithmetic dynamics, a subject well-suited for applications of the above techniques and results. Lastly, by work of Vojta and others, there exists a surprising correspondence between statements and theorems in Diophantine approximation and statements and theorems in Nevanlinna theory. The PI plans to study the analogues of the above problems in Nevanlinna theory, with the hope of enriching both subjects.