CAREER: Arithmetic, Algebraic, and Non-Archimedean Geometry

这是一个项目，将数论、代数、几何、拓扑和组合学的抽象和通常是基础的方面与算术、几何和经典数论中的具体和明确的问题联系起来。基本问题是寻找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。这些问题的简单和内在之美，以及解决这些问题的不成比例的深度和努力，自古希腊以来就激发了他们的研究。自然产生的核心问题如下：定性问题：存在解决方案吗？是否存在无限多的解决方案？解的集合是否有一些额外的结构(例如几何结构)？量化问题：有多少种解决方案？最小的解决方案有多大？我们如何才能明确而确定地找到所有的解决方案？隐含的问题：为什么方程有(或没有)解？为什么有些人有很多，而有些人什么都没有？是什么基本的数学结构控制着这一点？除了这项提议中的研究外，还开展了教育和外联活动，包括举办算术几何暑期研究生研究讲习班，以及继续开发开放、协作的MathOverflow网站。这一提议旨在用不太具体的、通常是基础领域的工具(例如，非阿基米德几何、代数堆栈、非阿贝尔方法、完美拟阵)来解决上述问题，目的是证明关于广义费马方程的新结果；开发技术以确定地找到给定方程的所有解；开发技术以限制给定方程的解的数目；以及建立新的工具(几何/上同调)来研究方程的解的结构。

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.