Analytic and Algebraic Methods in Discrete Geometry

该奖项支持PI在建立数学工具以解决离散几何中的组合问题方面的研究。离散几何研究基本的几何对象，如点、线和圆，以及它们的组合性质。因此，离散几何中的大多数组合问题都是非常直观的，可以用一种简单的方式来表示：覆盖平面上一个单位圆盘的条带的最小总宽度是多少？或者说，经过原点的直线中以相同角度分开的最大直线数是多少？与这些问题的简单表象相反，其中许多问题，包括本项目所考虑的问题，在很长一段时间内仍未得到解决，或者在作出巨大努力后直到最近才得到部分解决。该项目旨在进一步开发可用方法的工具箱，其中大多数是解析或代数性质的。同时，该项目将这些研究问题和主题整合到从高中到研究生的教育和推广活动中。这个项目的第一个主题是关于塔斯基木板问题的扩展和变化：如何有效地用木板覆盖平面上的凸体？塔斯基平板问题的推广是几何分析和凸几何的核心，并继续引起对凸体覆盖的几何和解析方面的兴趣。第二个主题涉及等角线——通过原点的线的集合，这些线以相同的角度成对分开。对等角线问题的研究取得了丰硕的成果，揭示了代数图论中许多具有挑战性的问题。一些问题的解决方案在运筹学、量子计算和通信理论中具有实际意义。以前的基础工作，包括PI所做的一些工作，已经表明来自拉姆齐理论、极值组合学、概率方法、谱图理论、代数组合学、拓扑组合学和代数几何的工具和见解可以帮助解决这个项目中的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports the PI's research in building mathematical tools to solve combinatorial problems in Discrete Geometry. Discrete geometry studies fundamental geometric objects, such as points, lines and circles, and their combinatorial properties. Thus most combinatorial problems in Discrete Geometry are quite visual and can be presented in a simple manner: What is the minimum total width of strips that cover a unit disk in the plane? Or, what is the maximum number of lines through the origin pairwise separated by the same angle? In contrast to the simple appearances of these problems, many of these problems, including the ones considered in this project, remain unsolved for an extended period or have been partly solved only recently following great efforts. This project aims to further develop the toolbox of available approaches, most of which are analytic or algebraic in nature. Simultaneously, the project integrates these research problems and themes into educational and outreach activities that extend from the high school level to the graduate.The first topic of this project concerns extensions and variations of Tarski's plank problem: How to efficiently cover a convex body in the plane using planks? Generalizations of Tarski’s plank problem are central to geometric analysis and convex geometry, and continue to generate interest in the geometric and analytic aspects of coverings of a convex body. The second topic concerns equiangular lines —- a collection of lines through the origin pairwise separated by the same angle. Investigation of the equiangular lines problem turns out to be fruitful and unearthed many challenging problems in algebraic graph theory. Solutions to some problems have practical consequences in Operational Research, Quantum Computation and Communication Theory. Previous foundational work, including some done by the PI, has shown that tools and insights from Ramsey Theory, Extremal Combinatorics, Probabilistic Methods, Spectral Graph Theory, Algebraic Combinatorics, Topological Combinatorics, and Algebraic Geometry can be helpful in solving the problems in this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.