CAREER: Arithmetic, Algebraic, and Non-Archimedean Geometry

职业：算术、代数和非阿基米德几何

• 批准号: 1555048
• 负责人: David Zureick-Brown
• 金额: $ 41.7万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1555048&HistoricalAwards=false
• 关键词: CAREER; Arithmetic; Algebraic; Non; Archimedean

This is a project to bring abstract and often foundational aspects of number theory, algebra, geometry, topology, and combinatorics to bear on concrete and explicit questions in arithmetic, geometry, and classical number theory. The basic problem is to find or describe all integer solutions of polynomial equations that arise in mathematics, especially cryptography, and the physical sciences. The simplicity and intrinsic beauty of these problems, and the disproportionate depth and effort of their resolution, has inspired their study since ancient Greece. The central questions that arise naturally stratify as follows: Qualitative questions: Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g. geometric)? Quantitative questions: How many solutions are there? How large is the smallest solution? How can we explicitly and with certainty find all solutions? Implicit questions: Why do equations have (or fail to have) solutions? Why do some have many and some have none? What underlying mathematical structures control this? The research in this proposal is complemented by educational and outreach activities, including the creation of a summer graduate research workshop on arithmetic geometry and the continued development of the open, collaborative MathOverflow web site. This proposal aims to attack the problems described above with tools from less concrete, often foundational fields (e.g., non-archimedan geometry, algebraic stacks, non-abelian methods, perfectoids), with the goals of proving new results on the generalized Fermat equation; developing techniques to find, with certainty, all solutions to a given equation; developing techniques to bound the number of solutions to a given equation; and building new tools (geometric/cohomological) to study the structure of solutions of equations.

这是一个项目，使抽象的，往往是数论，代数，几何，拓扑学和组合学的基础方面承担在算术，几何和经典数论的具体和明确的问题。基本问题是找到或描述数学中出现的多项式方程的所有整数解，特别是密码学和物理科学。这些问题的简单性和内在美，以及解决这些问题的不成比例的深度和努力，从古希腊开始就激发了他们的研究。自然产生的中心问题如下：定性问题：是否存在解决方案？ 是否存在无穷多个解？解的集合是否有一些额外的结构（例如几何结构）？定量问题：有多少种解决方案？最小解有多大？我们如何才能明确而有把握地找到所有的解决办法？隐含问题：为什么方程有（或没有）解？ 为什么有的人有很多，有的人没有？ 是什么基本的数学结构控制着这一切？这项建议中的研究得到了教育和外联活动的补充，包括设立一个关于算术几何的夏季研究生研究讲习班，以及继续开发开放、协作的MathOverflow网站。该提案旨在利用来自不太具体的、通常是基础性领域的工具（例如，非阿基米德几何，代数堆栈，非阿贝尔方法，perfectoids），与证明新的结果的目标广义费马方程;发展技术，找到，确定性，所有解决方案，以一个给定的方程;发展技术，以限制解决方案的数量，以一个给定的方程;和建立新的工具（几何/上同调），以研究方程的解决方案的结构。