Probabilistic and Extremal Combinatorics

概率和极值组合学

• 批准号: 2246907
• 负责人: Tom Bohman
• 金额: $ 24万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246907&HistoricalAwards=false
• 关键词: Probabilistic; Extremal; Combinatorics

This research project is an investigation of discrete mathematical objects like networks and codes. Extremal combinatorics is focused on developing a better understanding of discrete mathematical objects that optimize interesting or desirable properties, while probabilistic combinatorics studies discrete mathematical objects that are generated by a series of random choices. These two research directions are intimately related as randomized algorithms are a remarkably powerful tool for the construction of interesting discrete mathematical objects. Furthermore, randomness is a major theme of the work on extremal problems for discrete structures as a deep understanding of the ways in which deterministic objects mimic their randomized counterparts often leads to major progress. This research has the potential to benefit society through the development of new algorithms for computational problems on large networks, new methods for analyzing existing network algorithms, and new codes and communication protocols. The project also provides training opportunities at both the undergraduate and graduate level.This research is in the broad areas of probabilistic and extremal combinatorics. The work in probabilistic combinatorics is focused on very sharp concentration of global parameters of the binomial random graph and problems regarding the decomposition of the edge set of the uniform random graph into cliques or bicliques. The work on sharp concentration in the binomial random graph is motivated by a recent result of the investigator and a doctoral student that establishes 2-point concentration of the independence number of the binomial random graph over a broad range of the probability parameter. The comprehensive understanding of the extent of concentration of the independence number of the binomial random graph is one of the goals of this part of the research program. This project also includes further study of the fascinating lonely runner conjecture and some problems on Ramsey numbers for hypergraphs and posets. While the structures that we consider in these two contexts are not necessarily random, we expect the interplay of structure and randomness (i.e. pseudorandom properties of discrete structures) to play a major role. The ultimate goal of this research is to find new methods that are broadly applicable in discrete mathematics.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.

该研究项目是对网络和代码等离散数学对象的研究。极值组合学专注于更好地理解离散数学对象，优化有趣或理想的属性，而概率组合学研究由一系列随机选择生成的离散数学对象。这两个研究方向密切相关，因为随机算法是构建有趣的离散数学对象的非常强大的工具。此外，随机性是离散结构极值问题工作的一个主要主题，因为对确定性对象模仿其随机对应对象的方式的深入理解通常会带来重大进展。这项研究有可能通过开发大型网络计算问题的新算法、分析现有网络算法的新方法以及新的代码和通信协议来造福社会。该项目还提供本科生和研究生水平的培训机会。这项研究涉及概率和极值组合学的广泛领域。概率组合学的工作重点是二项式随机图全局参数的非常集中以及有关将均匀随机图的边集分解为团或双团的问题。关于二项式随机图中锐集中的工作是由研究者和一名博士生最近的一项结果推动的，该结果在广泛的概率参数范围内建立了二项式随机图独立数的 2 点集中。全面了解二项式随机图独立数的集中程度是这部分研究计划的目标之一。该项目还包括进一步研究令人着迷的孤独跑步者猜想以及超图和偏序集的拉姆齐数的一些问题。虽然我们在这两种情况下考虑的结构不一定是随机的，但我们预计结构和随机性（即离散结构的伪随机特性）的相互作用将发挥主要作用。这项研究的最终目标是找到广泛适用于离散数学的新方法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。