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.

该项目着重于开发新的数学工具来解决离散几何形状中的开放问题。离散的几何形状是数学的一个分支，涉及分析几何对象集合中的结构，包括点，线和圆。该项目的中心方向是研究凸域（例如球）的覆盖和交点特性。 PI旨在找到离散几何形状和其他数学字段之间的新连接。本科生将作为该项目的一部分进行指导。在长远的长期以来，该项目旨在探索较高维护空间中凸面体的覆盖物和相交模式的组合特性。在短期内，重点在于解决离散几何和组合凸的界面上的基本问题，包括覆盖问题的木板和Tverberg型问题。前者的方向介绍了塔斯基木板的各种变化，涵盖了猜想，例如仿射，多项式，球形和双曲线。第二个方向专用于Tverberg图，以及彩色和双重TVERBERG的猜想。为解决这些问题，PI计划通过离散和凸几何形状，组合代数，线性代数，功能分析和代数拓扑来应用和进一步开发方法，并特别着眼于优化技术。该奖项反映了NSF的法定任务，并通过对基础的智力效果进行评估，以评估依据。