Discrete Geometry and Convexity

离散几何和凸性

• 批准号: 2349045
• 负责人: Alexander Polyanskii
• 金额: $ 19.5万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-04-01 至 2027-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349045&HistoricalAwards=false
• 关键词: Discrete; Geometry; Convexity

This project focuses on developing new mathematical tools to address open problems in discrete geometry. Discrete geometry, as a branch of mathematics, involves analyzing structures within sets of geometric objects, including points, lines, and circles. The central direction of this project is the study of covering and intersection properties of convex domains such as balls. The PI aims to find new connections between discrete geometry and other mathematical fields. Undergraduate students will be mentored as part of this project.In the long term, this project aims to explore combinatorial properties of coverings and intersection patterns of convex bodies in higher-dimensional spaces. In the short term, the focus lies on addressing fundamental problems at the interface of discrete geometry and combinatorial convexity, including plank covering problems and Tverberg-type problems. The former direction addresses various variations of Tarski's plank covering conjecture such as affine, polynomial, spherical, and hyperbolic. The second direction is devoted to Tverberg graphs, as well as the colorful and dual Tverberg conjectures. To settle these problems, the PI plans to apply and further develop methods from discrete and convex geometry, combinatorics, linear algebra, functional analysis, and algebraic topology, with a particular focus on optimization techniques.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.

这个项目的重点是开发新的数学工具，以解决离散几何中的开放问题。离散几何作为数学的一个分支，涉及分析几何对象集（包括点、线和圆）内的结构。本项目的中心方向是研究球等凸域的覆盖和交性质。PI的目标是找到离散几何和其他数学领域之间的新联系。本科生将作为该项目的一部分进行指导。从长远来看，该项目旨在探索高维空间中凸体的覆盖和相交模式的组合性质。在短期内，重点在于解决离散几何和组合凸性界面上的基本问题，包括木板覆盖问题和Tverberg型问题。前一个方向解决了塔斯基的木板覆盖猜想的各种变化，如仿射，多项式，球面和双曲。第二个方向致力于Tverberg图，以及彩色和双Tverberg图。为了解决这些问题，PI计划应用和进一步发展离散和凸几何，组合学，线性代数，泛函分析和代数拓扑学的方法，特别关注优化技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。