Analytic and Combinatorial Problems in Number Theory

点击翻译按钮获取中文摘要

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.