CAREER: Ramsey Theory and Discrete Geometry

本研究计画探讨Ramsey理论与离散几何中的几个重要问题。 这些都是组合学和几何学中的问题，共同的主题是确定在某些限制下某个有限集可以有多大或多小。 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？ 其中许多问题已经吸引了数学家几十年，而另一些问题则是由计算几何的最新发展推动的。 PI将使用广泛的数学工具和技术来解决这些问题，包括概率、拓扑、代数几何和组合学。 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，并通过研究该领域中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学研究，并继续教授涵盖组合学和离散几何中最新结果和方法的课程。该项目的主要目标之一是获得经典超图Ramsey数的新界。 PI还将继续一系列最近的工作拉姆齐型问题的超图产生的几何，其中包括在更高的维度的埃尔多斯-Szekeres凸多边形问题，并找到大的独立集相交图的几何对象。 这个项目的第二个领域探讨了著名的Szemeredi-Trotter定理的各种扩展，以及它在加法数论中的应用。 具体的问题包括表征密集的点线安排，并估计在平面上的点和曲线之间的发病率。

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.