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Research in Extremal Combinatorics

极值组合学研究

基本信息

批准号：
2054452
负责人：
David Conlon
金额：
$ 30万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-04-01 至 2025-03-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054452&HistoricalAwards=false
关键词：
Research Extremal Combinatorics

项目摘要

Extremal combinatorics has 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 often significant breakthroughs, in number theory, design theory, probability, group theory, and theoretical computer science. The questions that are studied here represent obstacles to our understanding, and it is likely that progress on these challenges and the methods developed in solving them can later be exported to other areas. As such, a major goal of this project is to develop new tools and techniques and to expand on existing methods so that they apply in more general settings. This work will involve training undergraduate and graduate students.The PI will study a variety of fundamental questions in extremal combinatorics, spanning three main areas: extremal graph theory, Ramsey theory, and the study of pseudorandomness. In the first area, the PI intends to study several classical questions on extremal numbers, including the rational exponents conjecture and the compactness conjecture, as well as problems regarding the relationships between the homomorphism densities of different graphs. In the second area, the PI will tackle some of the classical problems on graph Ramsey theory, such as that of determining diagonal Ramsey numbers, as well as questions from arithmetic and geometric Ramsey theory. Finally, the PI intends to make further progress on developing the sparse regularity method and on finding and applying high-dimensional expanders. 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.

极值组合学在世纪在深度和广度上都有了显著的发展，产生了远远超出其原始设置的方法。这些包括应用，往往是重大突破，在数论，设计理论，概率论，群论和理论计算机科学。这里研究的问题是我们理解的障碍，在这些挑战方面取得的进展和为解决这些挑战而开发的方法以后可能会被推广到其他领域。因此，该项目的一个主要目标是开发新的工具和技术，并扩大现有的方法，使其适用于更一般的环境。这项工作将涉及培训本科生和研究生。PI将研究极值组合学中的各种基本问题，涵盖三个主要领域：极值图论、拉姆齐理论和伪随机性研究。在第一个领域，PI打算研究极值数的几个经典问题，包括有理指数猜想和紧性猜想，以及关于不同图的同态密度之间的关系的问题。在第二个领域，PI将解决一些关于图拉姆齐理论的经典问题，例如确定对角拉姆齐数，以及算术和几何拉姆齐理论的问题。最后，PI打算在发展稀疏正则性方法以及寻找和应用高维扩展器方面取得进一步的进展。共同的主题贯穿于这些领域，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的法定使命，并被认为值得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来支持。