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，从一个数据集，我们可以在平面上生成点或线的族。这引导我们研究几何集合有限族的结构。令人惊讶的是，该领域的许多突破都使用了来自等变拓扑的工具，乍一看，这似乎与最初的目标相去甚远。这个项目将加强计算几何、组合学和拓扑学之间的联系。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项目，为少数族裔学生准备夏季数学研究机会。该拓扑问题处理高维质量分配问题，如Stone-Tukey定理。该项目侧重于建立与一般公平划分问题（与无嫉妒划分有关）和与凸集相交模式有关的问题之间的联系。这些领域的关键结果都使用了类似的方法。这个项目将表明，上述相似性是所列问题家族之间更深层次联系的结果。这项工作将对如何利用拓扑工具来解决组合问题产生新的见解。该项目还包括对监督学习问题的计算应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。