CAREER: Algebraic Methods in Extremal Combinatorics

职业：极值组合中的代数方法

• 批准号: 1945200
• 负责人: Hao Huang
• 金额: $ 42.11万
• 依托单位: Emory 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: 2020-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945200&HistoricalAwards=false
• 关键词: CAREER; Algebraic; Methods; Extremal; Combinatorics

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.

极值组合学研究满足某些限制的组合对象的集合可以有多大或多小。这一数学分支在过去的几十年里经历了惊人的发展，并以其各种各样的方法和方法论成长为一个丰富的领域。这一奖项的主要目的是发展新的代数方法来解决极值组合问题，并进一步加深我们对图和超图的独立数和导出子结构的理解。这个项目涉及并旨在建立跨越许多领域的联系，包括代数、组合学、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暑期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工作的机会，并加强数学界和计算机科学界的研究合作。PI将学习几个基本的数学问题，包括：(I)对于极值组合数学的哪些结果，可以获得学位强化？(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)柯西的交错定理是否有一个量化版本？从这些项目中开发的技术将开启解决组合学中一些最重要和最具挑战性的公开问题的可能性：查瓦尔关于子族相交的猜想，托马谢夫斯基关于符号和的猜想，以及鄂尔多斯超图匹配猜想。这个奖项反映了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。

项目成果

On local Turán problems

关于图兰本地问题

• DOI: 10.1016/j.jcta.2020.105329
• 发表时间: 2021
• 期刊: Series A
• 作者: Frankl, Peter;Huang, Hao;Rödl, Vojtěch
• 通讯作者: Rödl, Vojtěch