On Regularity Methods and Applications in Graph Theory

这个数学研究项目集中在图论领域，这是一个活跃的组合学领域，由于它与数学和理论计算机科学的其他领域相联系，近年来取得了长足的进步。现代组合学中发展起来的许多工具，如正则性方法、概率方法和代数方法，在理解其他数学领域的问题时也被证明是有用的，例如数论和信息论。这个项目考虑了与图论相关的组合学中的几个基本问题。预计在这些问题上的进展将导致新的方法，不仅将在数学上产生影响，而且将在计算机科学中产生重要的实际应用。这个项目中探索的主题是组合学的中心问题之一。一个目标是通过理解几个重要应用中的界限来提高对正则性方法的能力和局限性的理解。另一个目标是确定使用概率方法的随机构造何时给出最优或接近最优界。几个经典话题包括Sidorenko猜想、Ramsey理论和二部图的图兰数。研究人员将使用并进一步开发多种技术来解决这些问题，包括Szmeredi正则性引理和弱正则性引理等正则性方法，以及图形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，被认为值得支持。

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.