CAREER: Discrete Geometry at the crossroads of Combinatorics and Topology

职业：组合学和拓扑学十字路口的离散几何

• 批准号: 2237324
• 负责人: Pablo Soberon Bravo
• 金额: $ 41.68万
• 依托单位: CUNY Baruch College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237324&HistoricalAwards=false
• 关键词: CAREER; Discrete; Geometry; crossroads; Combinatorics

Many computational and combinatorial problems can be reduced to problems in geometry. For example, from a data set, we can generate families of points or lines in the plane. This leads us to study the structure of finite families of geometric sets. Surprisingly, many breakthroughs in this area use tools from equivariant topology, which may, at first glance, seem far detached from the original goal. This project will reinforce the connections between computational geometry, combinatorics, and topology. Success in the objectives of this project will illuminate new connections between these areas. The project will also establish a program to prepare students from minoritized groups for summer research opportunities in mathematics. The topological problems of this proposal deal with high-dimensional mass partition problems, such as the Stone-Tukey theorem. The project focuses on establishing connections with general fair partition problems (related to envy-free partitions) and problems related to the intersection patterns of convex sets. These are areas in which crucial results use similar methods. This project will show that said similarity is a consequence of a deeper connection between the families of problems listed. This work will yield new insight into how one can leverage topological tools to solve combinatorial problems. The project also includes computational applications to problems in supervised learning.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.

许多计算和组合问题可以简化为几何问题。 例如，从数据集中，我们可以生成平面上的点族或线族。 这引导我们研究有限族几何集的结构。 令人惊讶的是，这一领域的许多突破都使用了等变拓扑的工具，乍一看，这似乎与最初的目标相去甚远。 该项目将加强计算几何、组合学和拓扑之间的联系。 该项目目标的成功将阐明这些领域之间的新联系。 该项目还将建立一个计划，为少数群体的学生提供暑期数学研究机会。该提案的拓扑问题涉及高维质量分配问题，例如斯通-图基定理。 该项目的重点是与一般公平划分问题（与无嫉妒划分相关）和与凸集相交模式相关的问题建立联系。 这些领域的关键结果都使用了类似的方法。 该项目将表明，上述相似性是所列出的问题系列之间更深层次联系的结果。 这项工作将为人们如何利用拓扑工具解决组合问题提供新的见解。 该项目还包括针对监督学习问题的计算应用。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。