Geometry in combinatorics

组合数学中的几何

• 批准号: 1301548
• 负责人: Boris Bukh
• 金额: $ 16万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301548&HistoricalAwards=false
• 关键词: Geometry; combinatorics

The project is about connections of combinatorics and geometry with the rest of mathematics. There are three main phenomena that the PI plans to investigate. The first is the appearance of rigid geometric structures as answers (or conjectured answers) to Turán- and Ramsey-type problems. Here, the goal is to understand why these structures appear, and to find a systematic way to go from a combinatorial problem to such a structure. The second phenomenon arises in the problems of approximation of large point sets. Here, the two sides of the problem lead to two different areas: the lower bounds to algebraic topology, and the upper bounds to logic. The PI will work on extending the existing techniques, with a goal of perfecting them to the point of eliminating the guesswork that they currently require. The final phenomenon concerns geometric incidence problems. The PI will try to generalize the existing topological approaches to finite field setting. The motivation here is the superficial similarity between the proofs in Euclidean space, such as the resolution of the distinct distance problem, and the proofs of sum-product-type results for dense sets in the finite field setting.The goal of this project is to strengthen the ties of combinatorics with areas of mathematics that are not commonly linked with combinatorics. Besides the immediate benefit of making progress on concrete combinatorial problems, the aim is to create bridges that will allow specialists in different fields to share insights. Furthermore, geometry is able to appeal to our imagination directly, and consequently can serve as a catalyst in combinatorial education.

该项目是关于组合和几何形状与其余数学的连接。 PI计划进行调查有三种主要现象。首先是刚性几何结构作为对Turán-和Ramsey型问题的答案（或猜想的答案）的出现。在这里，目标是了解为什么这些结构出现，并找到一种从组合问题到这种结构的系统方法。第二种现象是在大点集的近似问题中产生的。在这里，问题的两个方面导致了两个不同的区域：代数拓扑的下限以及上限到逻辑。 PI将致力于扩展现有技术，以使其完善它们，以消除他们当前需要的猜测。最终现象涉及几何事件。 PI将试图概括现有的拓扑方法以进行有限的现场设置。这里的动机是欧几里得空间中的证明之间的表面相似性，例如解决不同距离问题的解决方案，以及在有限田间设置中致密集的总成果型结果的证据。该项目的目的是加强与通常与组合局相关的数学领域的组合领域的联系。除了在具体组合问题上取得进展的直接好处外，目的是创建桥梁，使不同领域的专家可以共享见解。此外，几何形状能够直接出现我们的想象力，因此可以作为组合教育的催化剂。