FRG: Collaborative Research: Extremal Combinatorics and Flag Algebras

在极值组合学中，人们的目标是找到并刻画一类给定数学对象的最优成员。这些极端的物体通常具有独特的性质和结构，给出了许多数学领域的见解。在过去的二十年里，几种新技术在研究极端物体方面取得了非凡的成功，其中许多技术都是在计算机的帮助下进行的。最近最有效的方法之一是基于标志代数的理论。这种方法使研究人员能够将极值组合学问题转化为半定程序的实例，然后可以在计算机和学术以及商业软件的帮助下进行探索。这一翻译最近在长期悬而未决的问题上取得了突破。该方法具有很强的通用性，可应用于各种场景，如图、超图、排列、有向图、点集、嵌入图和系统发育树等。这个重点研究小组的目的是用这些技术解决ErdőS、图兰和扎兰凯维奇提出的三类突出的公开问题，这三类待研究的问题具有广义图兰问题的一般色彩，解决它们将产生深远的后果。第一类是极值超图问题，第二类是关于寻找具有某些性质的图的最大割的问题，第三类是与图的交叉数有关的问题。对于所有这三种情况，标志代数的使用最近都取得了重大进展，但并没有完全解决问题。该项目将把调查人员的专业知识与集中努力和进一步的方法开发结合起来，以解决这些悬而未决的问题。它计划找到与更传统的方法，如稳定性方法和线性代数方法的联系。大量的学生和早期职业研究人员将在这三个机构接受培训和支持，并计划举办几个有针对性的研讨会。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。

In extremal combinatorics one aims to find and characterize optimal members of a given class of mathematical objects. Those extremal objects often have unique properties and structures, giving insights in many mathematical fields. Over the last twenty years, several new techniques, many using the assistance of computers, have been extraordinarily successful in studying extremal objects. One of the recent powerful methods is based on the theory of flag algebras. This method enables researchers to translate extremal combinatorics questions into instances of semidefinite programs, which can then be explored with the help of a computer and academic as well as commercial software. This translation has led to recent breakthroughs on longstanding open questions. The method is versatile and can be applied in various settings such as graphs, hypergraphs, permutations, oriented graphs, point sets, embedded graphs, and phylogenetic trees. The aim of this focused research group is to resolve three prominent types of open questions by Erdős, Turán, and Zarankiewicz using these techniques.The three types of questions to be studied share the general flavor of generalized Turán problems, and solving them would have far reaching consequences. The first type are extremal hypergraph questions, the second type concerns finding maximum cuts in graphs with certain properties, and the third type are questions related to the crossing number of graphs. For all three, the use of flag algebras has recently led to significant progress but not to full solutions. This project will combine the expertise of the investigators with a concentrated effort and further method development to resolve these open questions. It is planned to find connections to more traditional methods such as the stability method and linear algebraic methods. A substantial number of students and early-career researchers will be trained and supported at the three institutions, and several focused workshops are planned.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.