CAREER: Problems in Extremal and Probabilistic Combinatorics

职业：极值和概率组合问题

• 批准号: 2146406
• 负责人: Asaf Ferber
• 金额: $ 43.5万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2146406&HistoricalAwards=false
• 关键词: CAREER; Problems; Extremal; Probabilistic; Combinatorics

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). In this project the PI will study various topics in extremal and probabilistic combinatorics, two areas which have grown significantly in both depth and breadth in the 21st century, resulting in methods that apply well beyond their original settings. These include applications, and many significant breakthroughs, in number theory, group theory, probability theory, information theory, and theoretical computer science. The approaches and techniques resulting from this project will have a significant impact on the development of these areas and will also be applicable in other branches of mathematics and theoretical computer science. This project is also designed for training undergraduate and graduate students.The problems the PI intends to study are fundamental and belong to some of the most actively studied topics of current research in extremal and probabilistic combinatorics. The first set of questions is coming from the sub-area of Ramsey theory and includes several classical questions on Ramsey numbers, spectral Ramsey theory, the clique number of Cayley graphs, Ramsey properties of random graphs, and more. The second set of questions is related to perfect matchings in (hyper)graphs. In particular, the PI will study fundamental problems such as: finding the Dirac threshold for the existence of a perfect matching, finding/counting 1-factorizations in (pseudo)random hypergraphs, decomposing the edges of d-regular (pseudo)random d-regular hypergraphs into perfect matchings, etc. Furthermore, the PI intends to study other interesting problems such as: finding the smallest number of (linear) bases over a finite field whose union forms an additive base, extremal problems in k-majority tournaments, counting Hadamard matrices, and more. Common themes run through these areas, and the methods developed in one area are likely to have implications for the others.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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。在该项目中，PI将研究极端和概率组合学的各种主题，这两个领域在21世纪的深度和广度都显着增长，从而导致方法远远超出了其原始环境。这些包括应用程序以及许多重大突破，在数量理论，群体理论，概率理论，信息理论和理论计算机科学上。该项目产生的方法和技术将对这些领域的发展产生重大影响，也将适用于数学和理论计算机科学的其他分支。该项目还旨在培训本科生和研究生。PI打算研究的问题是基本的，并且属于一些最积极研究的极端和概率组合学研究的主题。第一组问题来自拉姆西理论的子区域，包括有关拉姆西数字，光谱拉姆西理论，cayley图的集团数量，随机图的拉姆西属性等几个经典问题。第二组问题与（超级）图中的完美匹配有关。特别是，PI将研究基本问题，例如：找到完美匹配的越野毛阈值，在（伪）随机超图中找到/计数1个次数，将D-（pseudo）随机d-drorgular超图的边缘分解为完美的匹配等，例如，在较小的问题上，pi the linterive the Pie the Pie the Pie the Pie Intermore the Pie Intermore（pi）的数量很小。在一个有限的领域，该领域的工会在K-Mahority锦标赛中形成了加性基础，极端问题，计数Hadamard矩阵等等。在这些领域中运行的常见主题，在一个领域开发的方法可能会对其他领域产生影响。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准通过评估来支持的。