On Regularity Methods and Applications in Graph Theory

论图论中的正则方法及其应用

• 批准号: 1953958
• 负责人: Fan Wei
• 金额: $ 15万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953958&HistoricalAwards=false
• 关键词: Regularity; Methods; Applications; Graph; Theory

This mathematics research project centers on the area of graph theory, an active area of combinatorics that has made great strides in recent years because of its connection to other areas of mathematics and theoretical computer science. Many tools developed in modern combinatorics, such as the regularity methods, the probabilistic method, and algebraic methods, turn out to be also useful in understanding questions in other areas of mathematics such as number theory and information theory. This project considers several fundamental questions in combinatorics related to graph theory. It is expected that progress on these questions will lead to new methods that will have impact not only in mathematics but also in computer science, with important practical applications.The topics explored in this project are among the central questions in combinatorics. One goal is to improve understanding of the power and limitation of the regularity method through understanding the bounds in several important applications. Another goal is to determine when random constructions using the probabilistic method give optimal or nearly optimal bounds. Several classical topics include Sidorenko's conjecture, Ramsey theory, and Turan numbers of bipartite graphs. The investigator will use and further develop multiple techniques to tackle these problems, including regularity methods such as Szemeredi's regularity lemma and weak regularity lemmas, and analytic tools such as graph limits and random processes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该数学研究项目以图论领域为中心，这是组合数学的一个活跃领域，由于它与数学和理论计算机科学的其他领域的联系，近年来取得了长足的进步。现代组合学中开发的许多工具，例如正则性方法、概率方法和代数方法，对于理解数论和信息论等其他数学领域的问题也很有用。该项目考虑与图论相关的组合学中的几个基本问​​题。预计这些问题的进展将带来新的方法，这些方法不仅对数学产生影响，而且对计算机科学产生影响，并具有重要的实际应用。本项目探讨的主题是组合学的核心问题之一。目标之一是通过了解几个重要应用的界限来加深对正则性方法的威力和局限性的理解。另一个目标是确定使用概率方法的随机构造何时给出最佳或接近最佳边界。几个经典主题包括西多伦科猜想、拉姆齐理论和二分图的图兰数。研究者将使用并进一步开发多种技术来解决这些问题，包括 Szemeredi 正则性引理和弱正则性引理等正则性方法，以及图极限和随机过程等分析工具。该奖项反映了 NSF 的法定使命，并被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来提供支持。