CAREER: Ramsey Theory and Discrete Geometry

职业：拉姆齐理论和离散几何

• 批准号: 1651782
• 负责人: Andrew Suk
• 金额: $ 47.38万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2017-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1651782&HistoricalAwards=false
• 关键词: CAREER; Ramsey; Theory; Discrete; Geometry; CAREER; Ramsey; Theory; Discrete; Geometry

This research project examines several important problems in Ramsey theory and discrete geometry. These are problems arising from both combinatorics and geometry, with the common theme of determining how large or small a certain finite set can be under certain restrictions. For example, given a set of n points in the plane, how often can the unit distance occur among them? Many of these problems have fascinated mathematicians for decades, while others are fueled by the more recent development of computational geometry. The PI will tackle these problems using a wide range of mathematical tools and techniques from probability, topology, algebraic geometry, and combinatorics. One of the main goals of this project is to study the interplay between combinatorial and algebraic methods, and to further develop these methods by studying some of the most central open problems in the field. The PI will also continue to encourage high school, undergraduate, and graduate students to work in combinatorial research, and continue to teach courses which cover the latest results and methods used in combinatorics and discrete geometry.The first area in this project examines Ramsey-type problems in combinatorics and geometry. One of the major goals of this project is to obtain new bounds for classical hypergraph Ramsey numbers. The PI will also continue a sequence of recent works on Ramsey-type problems for hypergraphs arising in geometry, which include the Erdos-Szekeres convex polygon problem in higher dimensions, and finding large independent sets in intersection graphs of geometric objects. The second area of this project explores various extensions of the famous Szemeredi-Trotter theorem, and its applications in additive number theory. Specific problems include characterizing dense point-line arrangements, and estimating the number of incidences between points and curves in the plane.

该研究项目研究了拉姆齐理论和离散几何形状中的几个重要问题。 这些都是由组合学和几何形状引起的问题，其共同主题是确定在某些限制下可能有限的大小或小的有限集。 例如，给定平面中的一组n个点，单位距离一次可以在它们之间发生一次？ 这些问题中的许多人几十年来一直着迷数学家，而其他问题的计算几何形状的最新发展则助长了。 PI将使用概率，拓扑，代数几何和组合学的广泛的数学工具和技术来解决这些问题。 该项目的主要目标之一是研究组合方法和代数方法之间的相互作用，并通过研究该领域最中心的开放问题来进一步开发这些方法。 PI还将继续鼓励高中，本科生和研究生从事组合研究，并继续教授涵盖组合学和离散几何学的最新结果和方法的课程。该项目中的第一个领域研究了Ramsey-type型组合技术和几何学方面的问题。该项目的主要目标之一是为经典的Hypergraph Ramsey数字获得新的界限。 PI还将继续进行一系列有关在几何形状中引起的HyperGraphs问题的最新作品，其中包括Erdos-Szekeres在更高维度中的凸出多边形问题，并在几何学对象的相交图中找到大型独立集合。 该项目的第二个领域探讨了著名的szemeredi-trotter定理的各种扩展，及其在添加剂理论中的应用。 具体问题包括表征密集的点线排列，以及估计平面中点和曲线之间的发生率的数量。