Discrete Geometry and Extremal Combinatorics

离散几何和极值组合

• 批准号: 2246659
• 负责人: Andrei Pohoata
• 金额: $ 18.95万
• 依托单位: Emory University
• 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: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246659&HistoricalAwards=false
• 关键词: Discrete; Geometry; Extremal; Combinatorics

This project concerns several open problems at the intersection of discrete geometry and extremal combinatorics. Questions in discrete geometry traditionally involve sets of points, lines, triangles, planes, or other simple geometric objects, and many of them are tantalizingly natural and worth studying for their own sake. Some of them, such as the structure of 3-dimensional convex polytopes, go back to the antiquity, while others are also intimately connected with various different areas of modern mathematics, in particular extremal combinatorics. In recent years, these rich interactions have led to several remarkable developments between these two fields, and the goal of this project is to essentially capitalize as much as possible on this momentum. The first part of this project concerns Ramsey theory around the Erdős-Szekeres problem about the existence of large convex polytopes in finite configurations of point sets in general position, with an eye particularly towards establishing new upper bounds for various classical Ramsey numbers for graphs and hypergraphs. The second part of this proposal is about incidence geometry, an area with roots in Turán-type problems in extremal graph theory which is also fundamentally connected with other branches of mathematics, such as harmonic analysis and number theory, via the so-called sum-product phenomenon. The PI intends to further develop these connections by studying several old and new natural problems that arise on the different sides of this story. Examples of motivating (longstanding) questions include: the Zarankiewicz problem, the unit distance conjecture, and the Heilbronn triangle problem. As a byproduct, the PI also plans to develop new tools that could further the interplay between algebraic, analytic, combinatorial, and probabilistic methods in discrete mathematics. The PI plans to involve graduate students 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.

这个项目涉及离散几何和极值组合数学交叉的几个公开问题。离散几何中的问题传统上涉及一组点、直线、三角形、平面或其他简单的几何对象，其中许多问题非常自然，值得研究。它们中的一些，如三维凸多面体的结构，可以追溯到古代，而另一些也与现代数学的各个不同领域密切相关，特别是极值组合数学。近年来，这些丰富的互动导致了这两个领域之间的几个显著发展，该项目的目标是从本质上尽可能地利用这种势头。这个项目的第一部分是围绕ErdőS-Szekeres问题的Ramsey理论，该问题是关于在一般位置的点集的有限配置中存在大型凸多面体的问题，特别是着眼于建立图和超图的各种经典Ramsey数的新的上界。这一建议的第二部分是关于关联几何的，这是一个根植于极值图论中图兰型问题的领域，它也通过所谓的和积现象与调和分析和数论等其他数学分支从根本上联系在一起。PI打算通过研究这个故事不同方面出现的几个新旧自然问题来进一步发展这些联系。激励(长期存在的)问题的例子包括：Zarankiewicz问题、单位距离猜想和海尔布伦三角问题。作为副产品，PI还计划开发新的工具，以进一步促进离散数学中代数、分析、组合和概率方法之间的相互作用。PI计划让研究生参与这个项目。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。