Analytic and Combinatorial Problems in Number Theory

数论是研究整数的学科。数论中的一个中心概念是素数（即，只能被自己和一整除的数）。这些是整数的基本组成部分。虽然最初是出于纯粹的美学原因进行研究，但几十年来，数论和素数已经在计算机科学中找到了实际应用（例如，编码理论）和密码学。因此，素数在真实的生活中是无处不在的（例如，在密码系统中对通过因特网发送的消息进行编码）。本计画的研究是针对数论中的一些问题，使用分析与组合学的工具。这一领域也被称为算术组合学，近年来由于与数学的其他分支以及理论计算机科学的联系而获得了巨大的发展。本项目将探索与整数和素数的稠密子集有关的配置。由于Szemeredi、Furstenberg、Gowers、绿色和Tao等人的见解，已经取得了许多进展，但仍然有许多问题需要回答，还有许多有趣的配置需要探索。此外，由于张，梅纳德和陶最近在孪生素数猜想上的突破，使人们对素数有了更多的了解，PI打算在几乎孪生素数中找到模式。他还将研究函数场模型，其中有更多的工具，但也出现了额外的并发症。例如，由于Weyl差分在低特征下的失效，对函数域中高次多项式的分布的理解仍然是不完整的。

Number theory is the study of integers. One central notion in number theory is that of prime numbers (i.e., numbers divisible only by themselves and one). These are the building blocks of the integers. While originally studied for purely aesthetic reasons, for several decades number theory and prime numbers have found practical applications in computer science (e.g., in coding theory) and cryptography. As such, prime numbers are ubiquitous in real life (e.g., in cryptosystems to encode messages sent over the Internet). The research in this project is directed at some problems in number theory using tools from analysis and combinatorics. This area, also known as arithmetic combinatorics, has seen tremendous growth in recent years due to its connection with other branches of mathematics as well as theoretical computer science.This project will explore configurations pertaining to dense subsets of the integers and the primes. Thanks to the insights of Szemeredi, Furstenberg, Gowers, Green and Tao, among others, many advances have been made, but there are still many questions to answer and interesting configurations to explore. Furthermore, recent breakthroughs on the twin prime conjecture due to Zhang, Maynard, and Tao give more insights on the primes and the PI intends to find patterns in almost twin primes. He will also study the function field model, in which one has more tools but where additional complications also arise. For example, the understanding of the distribution of high-degree polynomials in function fields is still incomplete due to the failure of Weyl differencing in low characteristic.