Combinatorial Properties of Convex Sets and Measures in Euclidean spaces

欧几里得空间中凸集和测度的组合性质

• 批准号: 1764237
• 负责人: Pablo Soberon Bravo
• 金额: $ 10.52万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2018-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764237&HistoricalAwards=false
• 关键词: Combinatorial; Properties; Convex; Sets; Measures

Discrete geometry and topological combinatorics are important areas of mathematics. They highlight connections between different areas of mathematics in novel ways. The research in these subjects has applications in computer science, such as in search algorithms and optimization. This award will support the PI's research to develop new mathematical methods to solve problems on the boundary of combinatorics and topology. Problems in discrete geometry are often easy to state, and a good subject for popularization of mathematics. The PI will aim at involving undergraduate students in his research. The research of this project will focus on three main types of open problems. These type of problems highlight different aspects of discrete geometry and are multi-disciplinary; involving combinatorics, topology and linear algebra. As they are closely related, progress in one area will benefit the research in the others. The first kind of problems, partitions of measures, is a cornerstone of the interaction of equivariant topology and discrete geometry. The PI will focus on the effect of additional geometric constraints to classic problems in the field. The second is Tverberg theory, which focuses on understanding the convex hulls of finite sets of points from a combinatorial point of view. The PI has extended the linear-algebraic methods used for this area, and aims at working on topological versions of recent results. The third kind focuses on understanding the intersection structure of finite families of convex sets in Euclidean spaces. The work will focus on continuing a recent trend of developing quantitative versions of classic results, such as Helly-type theorems. Several of these problems have as additional motivation applications in computer science and other areas of mathematics.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.

离散的几何形状和拓扑组合学是数学的重要领域。 他们以新颖的方式突出了数学不同领域之间的联系。 这些主题的研究在计算机科学中具有应用，例如搜索算法和优化。 该奖项将支持PI的研究，以开发新的数学方法，以解决组合学和拓扑边界的问题。 离散几何形状的问题通常很容易说明，也是数学普及的好主题。 PI将旨在让大学生参与研究。该项目的研究将集中在三种主要类型的开放问题上。 这些类型的问题突出了离散几何形状的不同方面，并且是多学科的。涉及组合学，拓扑和线性代数。 由于它们与密切相关，一个领域的进步将使其他研究受益。 第一种问题，即测量的分区，是拓扑拓扑和离散几何形状相互作用的基石。 PI将重点介绍对现场经典问题的其他几何约束的影响。 第二个是Tverberg理论，它的重点是从组合的角度理解有限点的凸面。 PI扩展了用于该区域的线性偏格方法，旨在致力于最新结果的拓扑版本。 第三类着重于理解欧几里得空间中凸的有限家族的交点。 这项工作将着重于继续开发经典结果的定量版本的趋势，例如Helly-type定理。 这些问题中有几个在计算机科学和其他数学领域中具有其他动机应用。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。