Extremal Combinatorics: Themes and Challenging Problems

组合学是数学的一个基本领域。这个项目主要涉及图论领域，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的联系，近年来得到了蓬勃发展。许多图论问题也有实际的动机。世界上的大多数地方可以表示为由节点和某些节点对之间的连接组成的大型网络。例如，像Facebook这样的社交网络有超过20亿用户作为节点，友谊关系作为连接；像大脑这样的生物网络有超过1000亿个神经元作为节点，突触作为连接。了解这些网络并在其中设计快速算法具有很大的实用价值，例如了解新闻如何在社交网络中传播，了解大脑功能或疾病，以及为机器学习应用改进人工神经网络。这个项目考虑极值图论中的几个基本问题。该项目还为研究生和本科生提供了培训机会。PI计划使用并进一步发展多种技术，包括正则性方法，如Szmeredi正则性引理和弱正则性引理；分析工具，如图极限、随机过程和熵方法；以及各种其他组合工具。第一个项目与Szmeredi的正则性引理有关，该引理是现代图论中一个极其强大的工具，促使我们看待和研究图形的方式发生了戏剧性的变化。研究正则性引理的哪些应用具有更好的界是一个主要的研究方向。PI将研究几个经典问题，其目标是通过理解各种重要应用中的界限来提高我们对正则性方法的能力和局限性的理解。另一个主要项目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。几个经典主题包括Sidorenko猜想、Ramsey理论和图极限中的相关问题。目的是通过研究几个密切相关的具体问题，更好地理解这一大方向，并更深入地了解这些主题之间的联系。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。

Combinatorics is a fundamental area of mathematics. This project mainly concerns the area of graph theory, an active area of combinatorics which has been booming in recent years because of its connection to other areas of mathematics and theoretical computer science. Many graph theory problems also have practical motivations. Most of the world can be represented as large networks consisting of nodes and the connections between certain pairs of them. For example, a social network such as Facebook has over 2 billion users as nodes and friendship relations as connections; a biological network like the brain has over 100 billion neurons as nodes and synapses as connections. Understanding those networks and designing fast algorithms on them provides much practical value, examples include understanding how news spreads in a social network, understanding brain functions or diseases and improving artificial neural networks for machine learning applications. This project considers several fundamental questions in extremal graph theory. The project also provides training opportunities for graduate and undergraduate students.There are multiple techniques the PI plans to use and further develop, including regularity methods such as Szemeredi's regularity lemma and weak regularity lemmas; analytic tools such as graph limits, random processes and entropy methods; and various other combinatorial tools. The first project is related to Szemeredi's regularity lemma, which is an extremely powerful tool in modern graph theory which spurred a dramatic change of how we view and study graphs. It is a major direction of research to study which applications of the regularity lemma have considerably better bounds. The PI will work on several classical questions where the goal is to improve our understanding of the power and limitation of the regularity method through understanding the bounds in various important applications. Another major project 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 related questions in graph limits. The goal is to better understand this general direction through studying several closely related and concrete problems and gain more insight on the connections between these topics.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.