Problems in Combinatorial Geometry and Ramsey Theory

组合几何和拉姆齐理论中的问题

• 批准号: 2246847
• 负责人: Andrew Suk
• 金额: $ 25.18万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246847&HistoricalAwards=false
• 关键词: Problems; Combinatorial; Geometry; Ramsey; Theory

This research project studies several fundamental problems in combinatorial geometry and Ramsey theory. Combinatorial geometry is the study of extremal configurations of points, lines, and other simple geometric objects in Euclidean space. Understanding these extremal configurations is a fundamental mathematical problem, which also has several practical applications such as in motion planning in robotics, visibility and intersection problems in computer graphics, and frequency assignment problems in cellular networks. The area has seen tremendous growth over the past 15 years, with numerous unexpected connections to other mathematical areas such as number theory, logic, and computer science. One of the main goals of this project is to further explore these connections, and the interplay of methods from combinatorics (regularity lemma, probabilistic method, and the container method), topology (cell decomposition), algebraic geometry (polynomial method), and computer science (coding theory). Graduate students will be involved in this project.There are three main areas under investigation. The first area is in incidence geometry, and one of the major goals of this project is to characterize dense point-line arrangements in the plane. The second area of research is in the study of combinatorial and topological problems involving planar arrangements of curves, including graph drawings. The third area is in Ramsey theory, which is a fundamental area of combinatorics that focuses on the appearance of a specific configuration in a sufficiently large system. The PI will continue his long-term study of estimating classical graph and hypergraph Ramsey numbers. He will also study Ramsey-type problems with a geometric flavor, which involve point sets in general position, Heilbronn’s triangle problem, visibility graphs, and unavoidable crossing patterns in graph drawings.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.

该研究项目研究了组合几何学和拉姆西理论中的几个基本问​​题。组合几何形状是对欧几里得空间中点，线条和其他简单几何对象的极端配置的研究。了解这些极端配置是一个基本的数学问题，它在机器人技术，可见性和计算机图形中的相交问题中还具有多种实际应用，以及蜂窝网络中的频率分配问题。在过去的15年中，该领域的增长巨大，与其他数学领域（例如数字理论，逻辑和计算机科学）有许多意外的联系。该项目的主要目标之一是进一步探索这些连接，以及组合学（规则性引理，概率方法和容器方法）的方法相互作用，拓扑（细胞分解），代数几何（多条件方法）和计算机科学（编码理论）。研究生将参与该项目。正在调查三个主要领域。第一个区域是发病率的几何形状，该项目的主要目标之一是表征飞机上密集的点线布置。研究的第二个领域是研究组合和拓扑问题的研究涉及曲线的平面布置，包括图形图。第三个领域是拉姆西理论，这是组合制剂的一个基本领域，重点是在足够大的系统中特定配置的外观。 PI将继续他的长期研究估计经典图形和HyperGraph Ramsey数字。他还将以几何风味研究拉姆齐型问题，其中涉及一般位置的点集，海尔布隆的三角形问题，可见性图和不可避免的跨越图形上的交叉模式。该奖项反映了NSF的法定任务，并认为通过基金会的知识优点和广泛的范围来评估通过评估来获得评估，以此为宝贵。