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Super-Approximation in Number Theory

数论中的超近似

基本信息

批准号：
1602137
负责人：
Alireza Golsefidy
金额：
$ 23.82万
依托单位：
University of California-San Diego
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：
2016-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1602137&HistoricalAwards=false
关键词：
Super Approximation Number Theory

项目摘要

This research project concerns work related to questions in group theory, number theory, and analysis with application to cyber security. Being able to provide points that look random, statistically, is extremely important both from applied and pure points of view. An illustration of the importance of the task is given by secure communications, such as bank transfers and other electronic commerce, which start with a scheme that picks a random pair of prime numbers from the collection of all primes of an appropriate size. Being sufficiently random is one of the requirements that keeps many such schemes secure. However, providing a random point from a pool of points is a computational task that is much harder than it might seem. This research project studies mathematical questions underpinning the random choice of elements of a set. One of the main aims of this project is to establish that, if the group of symmetries of a collection of points is complicated enough, then one can provide a pseudo-random point rather quickly, compared to the number of points. The main focus of this project is to develop a very concrete way of measuring how complicated a group is by using polynomial equations. As part of the project, the investigator plans to explore some of its applications to pure mathematics, for example, generalizations of the work on sum and product sets that says: given a finite set of integers, the collection of all possible pairwise sums, and, the collection of all possible pairwise products, these collections cannot both be small. The principal aim of this project is to prove the main conjecture of super-approximation and explore its applications to the other areas of mathematics. Super-approximation roughly says: whether a random walk (with respect to finitely many elements with algebraic entries) in invertible matrices over some topological ring, e.g., real numbers, p-adic numbers, or adeles, has spectral gap depends only on the Zariski closure of the group generated by the given set of matrices. In the past decade there has been substantial progress in this subject, and super-approximation has been shown to be extremely useful in various parts of mathematics, including number theory (affine sieve; variations of Galois representations) and group theory (sieve in groups; hyperbolic geometry; orbit equivalence rigidity). Another goal of this project is to find additional applications of super-approximation.

该研究项目涉及与群论、数论和网络安全应用分析问题相关的工作。从应用和纯粹的角度来看，能够提供看起来随机的、统计上的点都是极其重要的。安全通信（例如银行转账和其他电子商务）说明了该任务的重要性，这些通信首先从一个方案开始，该方案从所有适当大小的素数的集合中挑选一对随机的素数。足够的随机性是保证许多此类方案安全的要求之一。然而，从点池中提供随机点是一项计算任务，比看起来要困难得多。该研究项目研究支持随机选择集合元素的数学问题。该项目的主要目标之一是确定，如果一组点的对称性足够复杂，那么与点的数量相比，我们可以相当快地提供伪随机点。该项目的主要重点是开发一种非常具体的方法，通过使用多项式方程来测量群的复杂程度。作为该项目的一部分，研究人员计划探索其在纯数学中的一些应用，例如，对和集和乘积集的工作的概括：给定一个有限的整数集，所有可能的成对和的集合，以及所有可能的成对乘积的集合，这些集合不能都很小。该项目的主要目的是证明超近似的主要猜想并探索其在数学其他领域的应用。超近似粗略地说：在某些拓扑环上的可逆矩阵（例如实数、p进数或阿德尔）中的随机游走（相对于具有代数项的有限多个元素）是否具有谱间隙仅取决于由给定矩阵集生成的群的 Zariski 闭包。在过去的十年中，这一学科取得了实质性进展，超近似已被证明在数学的各个部分都非常有用，包括数论（仿射筛；伽罗瓦表示的变体）和群论（群筛；双曲几何；轨道等效刚度）。该项目的另一个目标是找到超近似的其他应用。