Modern Approaches for Classical Diophantine Problems

这个研究项目涉及丢番图逼近和数的几何。丢番图近似处理近似的真实的号码由有理数和问题的分类，给定的号码作为不合理的，代数，或超越。 丢番图方程是一组整数系数的代数方程，求解该方程组需要求整数解。数的几何学涉及使用几何概念来解决数论中的问题，通常通过整数方程的解。数学的基本问题之一是求给定多项式方程的解。这些问题的难度取决于多项式的形状，更取决于人们正在寻找什么样的解决方案。找到作为多项式方程的解的复数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，是数论的焦点。 本研究计划旨在扩阔及深化这一基础领域的知识，研究丢番图方程积分解的一般计数问题及其应用。PI将通过将古典分析技术与解析数论和算术几何的现代应用相结合来发展方法论的进步，以有效和明确地解决一些重要和长期存在的丢番图问题。本文还研究了丢番图方程积分解的分布。该项目的成果预计将导致更好地理解某些算术对象的几何和分析特性，如椭圆曲线。

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.