CAREER: Algebraic Methods in Extremal Combinatorics

极值组合学研究满足一定限制条件的组合对象的集合可以有多大或多小。数学的这一分支在过去的几十年里得到了惊人的发展，并发展成为一个拥有各种各样自己的方法和方法的丰富领域。该奖项的主要重点是开发新的代数方法来解决极值组合问题，并进一步了解图和超图的独立数和诱导子结构。该项目涉及并旨在建立跨多个领域的联系，包括代数，组合数学，概率和离散几何。该项目的一个组成部分是其教育部分，其中包括组织初级研究讲习班和夏季REU计划。该奖项的长期教育目标是积极吸引本科生参与STEM研究，为早期职业研究人员提供宣传他们的作品的机会，并加强数学和计算机科学社区之间的研究合作。PI将研究几个基本的数学问题，包括：（i）对于极值组合学的哪些结果可以期待学位强化？(ii)图或超图的（伪）邻接矩阵的谱在多大程度上描述了图的独立数或导出子结构？(iii)柯西交错定理有定量的版本吗？这些项目开发的技术将为攻克组合数学中一些最重要和最具挑战性的开放问题提供可能性：Chvatal关于交叉子族的猜想，Tomaszewski关于符号和的猜想，以及Erdos超图匹配猜想。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

Extremal combinatorics studies how large or how small a collection of combinatorial objects satisfying certain restrictions can be. This branch of mathematics has witnessed spectacular development in the last few decades, and grown into a rich field with a wide variety of its own approaches and methodology. The main focus of this award is to develop new algebraic methods to solve extremal combinatorial problems, and further our understanding of the independence number and induced substructures of graphs and hypergraphs. This project involves and aims to establish connections across numerous areas, including algebra, combinatorics, probability, and discrete geometry. An integral part of this project is its educational component, which includes organizing junior research workshops and summer REU programs. The long-term education goal of this award is to actively engage undergraduate students in STEM research, provide opportunities for early-career researchers to publicize their works, and enhance the research collaboration between the Mathematics and Computer Science communities.The PI will study several fundamental mathematical questions, including: (i) For which results in extremal combinatorics one can expect a degree strenthening? (ii) To what extent the spectrum of the (pseudo-)adjacency matrix of a graph or hypergraph describes the independence number or induced substructures of a graph? (iii) Is there a quantitative version of Cauchy's Interlace Theorem? Techniques developed from these projects will open the possibility of attacking some of the most important and challenging open problems in combinatorics: Chvatal's Conjecture on intersecting subfamilies, Tomaszewski's Conjecture on signed sums, and the Erdos hypergraph matching conjecture.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.