Problems in the Geometry of Numbers and Diophantine Analysis

数学中的一个基本和中心问题，可以追溯到古代，涉及到多项式方程的整数解的理解。这个问题一直是现代数学的灵感来源，因为用于研究这个问题的技术帮助塑造了几个领域的基础，如数论、代数和分析。这项工作将通过研究各种多项式方程族和在数论的丰富领域中找到创新的应用，为该领域带来新的视角。该计划还将加强对研究生的培训，并为本科生提供指导活动和研究机会。主要研究者的目标是利用经典丢番图分析和解析数论和算术几何的现代应用工具来研究基本的算术结构。这项工作将把数字几何的思想与代数和分析工具结合起来，研究诸如用二进制形式表示整数和数域中的顺序等问题，以及关于数域中高度的各种问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A fundamental and central question in mathematics, going back to antiquity, concerns understanding integer solutions to polynomial equations. This question has been a source of inspiration in modern mathematics, as the techniques used to study this question have helped shape the foundation of several areas such as number theory, algebra, and analysis. This work will bring new perspectives to the field by studying various families of polynomial equations and by finding innovative applications in the rich area of number theory. The project will also enhance the training of graduate students and support mentoring activities and research opportunities for undergraduate students. The principal investigator aims to utilize tools from classical Diophantine analysis and modern applications of analytic number theory and arithmetic geometry to study basic arithmetic structures. This work will combine ideas from the geometry of numbers with both algebraic and analytic tools to study problems such as representation of integers by binary forms and orders in number fields, as well as various problems about heights in number fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.