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质数（即只能被自己和1整除的数）。这些是整数的组成部分。虽然最初纯粹是出于美学原因而研究，但几十年来，数论和素数在计算机科学（例如编码理论）和密码学中找到了实际应用。因此，质数在现实生活中无处不在（例如，在通过互联网发送的消息编码的密码系统中）。本课题利用分析学和组合学的工具研究数论中的一些问题。这个领域，也被称为算术组合学，近年来由于它与数学的其他分支以及理论计算机科学的联系而取得了巨大的增长。这个项目将探索关于整数和素数的密集子集的配置。由于Szemeredi， Furstenberg, Gowers， Green和Tao等人的见解，已经取得了许多进展，但仍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Zhang、Maynard和Tao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，但也会产生额外的复杂性。例如，由于低特征Weyl差分的失效，对函数场中高次多项式的分布的理解仍然不完整。

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.