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，我们可以在平面中生成点或线的家族。 这使我们研究了几何集的有限家族的结构。 出乎意料的是，该领域的许多突破都使用了Equivariant拓扑的工具，乍一看似乎与最初的目标相距甚远。 该项目将加强计算几何，组合和拓扑之间的联系。 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，以使少数小组的学生为数学研究机会做好准备。该提案的拓扑问题涉及高维群众分区问题，例如石键定理。 该项目着重于建立与一般公平分区问题（与嫉妒的分区有关）以及与凸集集合相交模式相关的问题。 这些是关键结果使用类似方法的领域。 该项目将表明，所述相似性是列出的问题家族之间更深层次联系的结果。 这项工作将为如何利用拓扑工具解决组合问题提供新的见解。 该项目还包括对监督学习问题的计算应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估的。