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

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.

该研究项目研究拉姆齐理论和离散几何中的几个重要问题。 这些问题都是由组合学和几何学产生的，其共同主题是确定某个有限集在某些限制下可以有多大或多小。 例如，给定平面上的一组 n 个点，单位距离在它们之间出现的频率是多少？ 其中许多问题几十年来一直让数学家着迷，而其他问题则受到计算几何的最新发展的推动。 PI 将使用概率、拓扑、代数几何和组合学等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，并通过研究该领域一些最核心的开放问题来进一步发展这些方法。 PI还将继续鼓励高中生、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中使用的最新成果和方法的课程。该项目的第一个领域研究拉姆齐型组合数学和几何问题。该项目的主要目标之一是获得经典超图拉姆齐数的新界限。 PI 还将继续开展一系列关于几何中出现的超图 Ramsey 型问题的最新工作，其中包括高维中的 Erdos-Szekeres 凸多边形问题，以及在几何对象的交图中查找大型独立集。 该项目的第二个领域探索著名的 Szemeredi-Trotter 定理的各种扩展及其在加法数论中的应用。 具体问题包括描述密集点线排列的特征，以及估计平面中点和曲线之间的重合数量。