CAREER: Integral Points on Varieties and Related Tools and Topics

理解多项式方程组的整数解集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的集合是否是有限的，如果是，如何显式地计算这样的解的集合。将强大的几何思想和概念引入这一学科，已经取得了重大进展，并对基本问题有了广泛而深入的理解。从这个几何观点出发，本研究项目旨在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。除了研究活动外，PI还将为密歇根州立大学的一组精选的本科生组织和发起学生和教师的研究项目。这些研究项目将通过本科生系列讨论会和资助本科生的会议旅行得到加强。PI将继续密切参与研究生教育，并指导研究生和博士后，提出与拟议研究方向相关的问题。PI将研究几个相互关联的问题，这些问题围绕着对变量上的积分点的研究。首先，PI将沿着PI的猜想，研究将曲线上积分点的西格尔定理推广到高维变量的问题。在第二个项目中，PI将研究曲线上有界度的积分点。然后，在不同的方向上，将研究有效计算曲线和高维变量上的积分点问题的各个方面。研究变异上的积分（或有理）点的基本工具来自丢番图近似。其中一个关键工具就是施密特子空间定理。PI将继续他最近的工作，证明施密特定理的推广，以代数点的施密特- wirsing型猜想为最终目标。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主题。最后，通过Vojta等人的工作，在丢芬图近似中的命题和定理与内万林纳理论中的命题和定理之间存在着惊人的对应关系。PI计划在奈万林纳理论中研究上述问题的类似问题，希望丰富这两个学科。

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.