Extremal Combinatorics

极值组合学

• 批准号: 1201380
• 负责人: Po-Shen Loh
• 金额: $ 30万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201380&HistoricalAwards=false
• 关键词: Extremal; Combinatorics

This project focuses on topics in Extremal Combinatorics, which investigate the relationships between useful parameters of discrete systems, and characterizes their extreme values over various families of those systems. Such problems often have applications in Computer Science and other areas of Mathematics, but are also elegant and interesting in their own right. The goal of this project is to further develop the toolbox of available approaches for attacking Combinatorial problems. This goal will be achieved by studying certain families of problems which can be organized according to the methods used in their solutions. First, we examine applications of analytical (continuous) arguments to discrete problems. Second, we study extremal problems related to set systems. Finally, we explore the application of probabilistic methods to purely deterministic problems.The field of combinatorics encompasses the rigorous mathematical study of discrete structures and processes, such as sets, networks, and even algorithms. In the past, combinatorial problems were often solved by pure ingenuity. Today, however, a variety of powerful methods have emerged, which draw elements from many other branches of the mathematical sciences. At the same time, the rapid development of Computer Science has substantially increased the demand for foundational results on discrete systems--these can provide the theoretical basis for future work. The overarching objective of this project is to use a problem-driven philosophy to inspire innovations in the development of new techniques in Combinatorics. Unifying the problems in this study is a theme of simple, elegant statements that inspire approaches which arise from across the mathematical spectrum. Consequently, these problems also serve well as a platform for bringing young people into research.

该项目重点关注极值组合学主题，研究离散系统的有用参数之间的关系，并表征它们在这些系统的各个系列中的极值。 此类问题通常在计算机科学和其他数学领域中得到应用，但其本身也很优雅且有趣。 该项目的目标是进一步开发解决组合问题的可用方法的工具箱。 这一目标将通过研究某些问题系列来实现，这些问题可以根据解决方案中使用的方法进行组织。首先，我们研究分析（连续）论证在离散问题中的应用。 其次，我们研究与集合系统相关的极值问题。 最后，我们探索概率方法在纯粹确定性问题中的应用。组合学领域涵盖离散结构和过程的严格数学研究，例如集合、网络，甚至算法。 过去，组合问题通常是靠纯粹的聪明才智来解决的。然而，如今，各种强大的方法已经出现，它们借鉴了数学科学许多其他分支的元素。 与此同时，计算机科学的快速发展极大地增加了对离散系统基础成果的需求——这些成果可以为未来的工作提供理论基础。 该项目的总体目标是利用问题驱动的理念来激发组合学新技术开发的创新。 统一本研究中的问题是一个简单、优雅的陈述的主题，激发了来自整个数学领域的方法。 因此，这些问题也可以很好地为年轻人提供研究的平台。