Geometric Ramsey Theory and Incidence Geometry

本课题研究的是处于组合学和几何学交界处的离散几何中的几个问题。该领域中的问题通常被简单地描述，并且通常涉及欧几里德空间中的基本对象，如点、线、球等。虽然离散几何中的许多问题本身就值得研究，但其他问题的动机是它们在计算机科学和工程中的应用。最近，离散几何有了巨大的发展，人们发现了许多与其他数学领域意想不到的联系。这个项目的主要目标之一是进一步探索这些联系，并应用新的工具和技术来解决几个长期悬而未决的问题。PI还将继续鼓励高中生、本科生和研究生从事组合研究，并继续教授涵盖该领域最新结果和公开问题的课程。目前正在调查的主要有两个领域：几何拉姆齐理论和关联几何。在过去的几十年里，Ramsey数被广泛地应用于几何中出现的齐性问题的上界。对于许多这样的应用，人们可以通过利用边是用代数定义的这一事实来获得更好的边界。PI将继续最近关于代数定义的超图的Ramsey型问题的一系列工作，包括著名的Erdos和Szekeres的幸福结局问题，以及在几何对象的交图中寻找大的独立集。这个项目的另一个主要目标是探索Szmeredi-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的点-线排列以及估计点和d维变量之间的关联数目。

This research project studies several problems in discrete geometry, which lies on the interface of combinatorics and geometry. Problems in the field are often simply described, and typically involve fundamental objects in Euclidean space such as points, lines, spheres, etc. While many questions in discrete geometry are worth studying for their own sake, others are motivated by their applications in computer science and engineering. More recently, discrete geometry has seen tremendous growth, and numerous unexpected connections to other fields of mathematics are being discovered. One of the main goals of this project is to further explore these connections, and to apply new tools and techniques to several long-standing open problems. The PI will also continue to encourage high-school, undergraduate, and graduate students to work in combinatorial research, and continue to teach courses that cover the latest results and open problems in the field.There are two main areas under investigation: geometric Ramsey theory and incidence geometry. Over the past few decades, Ramsey numbers have been applied extensively to give upper bounds on homogeneity problems arising in geometry. For many of these applications, one can obtain much better bounds by exploiting the fact that the edges are defined algebraically. The PI will continue a sequence of recent works on Ramsey-type problems for hypergraphs defined algebraically, which includes the famous Happy Ending Problem of Erdos and Szekeres, and finding large independent sets in intersection graphs of geometric objects. Another major goal of this project is to explore various extensions of the Szemeredi-Trotter theorem and its applications to additive number theory. Specific problems include characterizing dense point-line arrangements and estimating the number of incidences between points and d-dimensional varieties.