The Sunflower Conjecture, Disjunctive Normal Forms, and Beyond

向日葵猜想、析取范式及其他

• 批准号: 1953928
• 负责人: Shachar Lovett
• 金额: $ 30万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953928&HistoricalAwards=false
• 关键词: Sunflower; Conjecture; Disjunctive; Normal; Forms

There is a rich history of symbiosis between computer science and mathematics. This award is aimed at further strengthening this connection, and in particular between the fields of theoretical computer science and combinatorics. This project centers around a fundamental open problem in combinatorics, called the sunflower conjecture. This problem originated in 1960 and is still unanswered, despite having numerous applications in mathematics and in computer science. Recent work by the PI has made significant progress towards resolving the conjecture, and this project is focused on continuing this work. The project provides research training opportunities for graduate students.Concretely, the project explores a correspondence between set systems (also known as hypergraphs) and DNFs (Disjunctive Normal Forms). Building upon recent work by the PI, the project demonstrates how tools used to study DNFs can be instrumental in studying set systems, and proposes a unified and versatile framework to make progress on several well-studied conjectures in mathematics and theoretical computer science about the structure of set systems and DNFs. These also include, beyond the sunflower conjecture and related problems in combinatorics, problems related to DNF compression and the Fourier structure of DNFs.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.

计算机科学和数学之间的共生关系有着悠久的历史。该奖项旨在进一步加强这种联系，特别是理论计算机科学和组合学领域之间的联系。这个项目围绕组合学中一个基本的开放问题，称为向日葵猜想。这个问题起源于1960年，尽管在数学和计算机科学中有许多应用，但仍然没有答案。PI最近的工作在解决这个猜想方面取得了重大进展，这个项目的重点是继续这项工作。该项目为研究生提供了研究培训的机会。具体来说，该项目探索了集合系统（也称为超图）和DNF（析取范式）之间的对应关系。在PI最近工作的基础上，该项目展示了用于研究DNF的工具如何有助于研究集合系统，并提出了一个统一的通用框架，以在数学和理论计算机科学中关于集合系统和DNF结构的几个研究良好的架构上取得进展。除了向日葵猜想和组合学中的相关问题，还包括与DNF压缩和DNF的傅立叶结构相关的问题。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，被认为值得支持。

项目成果

Hypercontractivity on high dimensional expanders

• DOI: 10.1145/3519935.3520040
• 发表时间: 2021-11
• 期刊: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
• 作者: Mitali Bafna;Max Hopkins;T. Kaufman;Shachar Lovett

Fractional Certificates for Bounded Functions

有界函数的分数证明

• 发表时间: 2023
• 期刊: Leibniz international proceedings in informatics
• 作者: Lovett, Shachar;Zhang, Jiapeng
• 通讯作者: Zhang, Jiapeng

Sampling Equilibria: Fast No-Regret Learning in Structured Games

抽样均衡：结构化博弈中的快速无悔学习

• 发表时间: 2023
• 期刊: Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
• 作者: Beaglehole, Daniel;Hopkins, Max;Kane, Daniel;Liu, Sihan;Lovett, Shachar
• 通讯作者: Lovett, Shachar

High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

• DOI: 10.1137/1.9781611977073.47
• 发表时间: 2020-11
• 作者: Mitali Bafna;Max Hopkins;T. Kaufman;Shachar Lovett

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

• DOI: 10.48550/arxiv.2205.00644
• 发表时间: 2022-05
• 作者: J. Gaitonde;Max Hopkins;T. Kaufman;Shachar Lovett;Ruizhe Zhang