Topology and Linear Algebra in Discrete Geometry

离散几何中的拓扑和线性代数

基本信息

批准号：
2054419
负责人：
Pablo Soberon Bravo
金额：
$ 19.51万
依托单位：
CUNY Baruch College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054419&HistoricalAwards=false
关键词：
Topology Linear Algebra Discrete Geometry

项目摘要

Discrete geometry is the study of combinatorial properties of finite families of geometric objects. Results in this field often have applications to search algorithms, data classification, and optimization. The methods used to answer discrete geometry questions can be wildly different. Some are based on linear algebra, where the rigid structure of the space is important. Some are based on topology, and the results hold up to smooth deformations of the objects involved. This research project aims to develop new ways to use topological and linear algebraic methods in discrete geometry to expand the connections between combinatorics and other fields. The driving motivation is to understand what makes a combinatorial problem amenable to techniques from topology, and what makes it amenable to techniques from linear algebra. The project also includes topics that can be used to mentor undergraduate students in research. The project focuses on questions related to finite families of convex sets. The PI plans to develop new ways to apply topological methods to Tverberg-type problems, related to a longstanding "colorful" version of Tverberg’s theorem. The recent progress in quantitative Helly-type theorems can help build a bridge between the analytic side of convexity and linear programming, which the PI aims to formalize. Several of the questions under study aim to generalize classic results in extremal combinatorics to high-dimensional Euclidean spaces.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.

离散的几何形状是对几何对象有限家族的组合特性的研究。该领域的结果通常具有搜索算法，数据分类和优化的应用程序。用于回答离散几何问题的方法可能大不相同。有些基于线性代数，其中空间的刚性结构很重要。有些基于拓扑，结果得出了所涉及的对象的平滑变形。该研究项目旨在开发新的方法来使用离散几何形状中的拓扑和线性代数方法来扩展组合学和其他领域之间的联系。驱动动机是了解是什么使拓扑结构适合技术的组合问题，以及是什么使其适合线性代数的技术。该项目还包括可用于研究精神本科生的主题。该项目着重于与凸组有限家庭有关的问题。 PI计划开发新的方法，将拓扑方法应用于Tverberg型问题，这与Tverberg定理的长期“色彩”版本有关。定量HELLY型定理的最新进展可以帮助在凸的分析方面和线性编程之间建立桥梁，PI旨在正式化。研究中的几个问题旨在将极端组合学的经典成果推广到高维欧几里得空间。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响评估标准来评估，并通过评估来诚实地支持。