Modern Approaches for Classical Diophantine Problems

这个研究项目涉及丢番图近似和数的几何。丢芬图近似处理的是用有理数逼近实数的问题，以及将给定数分类为无理数、代数数或超越数的问题。丢芬图方程是系数为整数的代数方程，求其整数解。数字几何涉及使用几何概念来解决数论中的问题，通常是通过解决整数方程。数学的基本问题之一是求给定多项式方程的解。这类问题的难度取决于多项式的形状，更重要的是取决于我们在寻找什么样的解。找到复数作为多项式方程的解是一项相对容易的任务。然而，寻找多项式方程的整数解是一个更加微妙和深入的问题，是数论的重点。该项目旨在拓宽和深化这一基础领域的知识。本课题主要研究丢番图方程的积分解计数的一般问题及其应用。PI将通过将经典分析技术与解析数论和算术几何的现代应用相结合，有效而明确地解决一些重要而长期存在的丢番图问题，从而发展方法上的进步。研究丢番图方程的积分解的分布。预计该项目的结果将导致对某些算术对象（如椭圆曲线）的几何和解析性质的更好理解。

This research project concerns Diophantine approximation and the geometry of numbers. Diophantine approximation deals with approximation of real numbers by rational numbers and with questions of classification of given numbers as irrational, algebraic, or transcendental. Diophantine equations are algebraic equations with integer coefficients, for which integer solutions are sought. The geometry of numbers deals with the use of geometric notions to solve problems in number theory, usually via the solutions of equations in integers. One of the basic problems of mathematics is to find the solutions of a given polynomial equation. The level of difficulty of such problems depends on the shape of the polynomial and more so on what kind of solutions one is looking for. Finding complex numbers that are solutions to a polynomial equation is a relatively easy task. However, finding integer solutions to polynomial equations is a problem of much greater subtlety and depth, and is the focus of number theory. This project aims to broaden and deepen knowledge in this fundamental area.The purpose of this research project is to study the general problem of counting integral solutions of Diophantine equations and its applications. The PI will develop methodological advances by combining techniques from classical analysis with modern applications of analytic number theory and arithmetic geometry to address some important and long standing Diophantine problems effectively and explicitly. The research is also concerned with studying the distribution of integral solutions to Diophantine equations. Results of the project are anticipated to lead to substantially better understanding of the geometric and analytic properties of certain arithmetic objects, such as elliptic curves.