Topological Methods for Discrete Problems

离散问题的拓扑方法

• 批准号: 1855591
• 负责人: Florian Frick
• 金额: $ 18万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855591&HistoricalAwards=false
• 关键词: Topological; Methods; Discrete; Problems

We often encounter situations with inherent symmetries, such as attempting to fairly divide a good among several parties: If our division is truly fair, we could permute the pieces without changing the outcome. This viewpoint arises also in more abstract settings, such as decomposing a data set in a balanced way with linear equations to efficiently handle it, or evaluating a function in a fixed set of points to compute its integral. Note that if we introduce subjective preferences of the involved parties to our fair division problem, the inherent symmetries disappear. Again this loss of symmetry is a natural phenomenon for the class of the problems mentioned so far. This project develops methods to approach such problems, even in the absence of symmetry. Notably, the methods are topological, that is, continuous, whereas the problems have a combinatorial, or discrete, flavor. Thus this research builds bridges between different branches of mathematics, making problems in one branch amenable to the tools of another.The PI will conduct research on a selection of combinatorial problems that benefit from a geometric viewpoint. Among the themes of research are intersection patterns of sets, both in a purely combinatorial setting (such as hypergraph matchings and design theory) as well as in geometric settings (such as intersections of convex sets and non-embeddability of complexes into Euclidean space), and the study of structured sets in abelian groups (such as sumsets, the structure of sets containing zero-sums, and spherical designs). These are problems, where one is interested in constructing an object or showing its non-existence, but global obstructions emerge---local solutions do not glue together to form the desired object. Geometry and topology provide a suitable framework to organize these non-local phenomena and detect such obstructions in the large. This research focuses on problems that admit geometric generalizations and aims to delimit rigid discrete results from their more flexible geometric counterparts.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.

我们经常遇到具有内在对称性的情况，例如试图在几方之间公平地划分一种商品：如果我们的划分是真正公平的，我们可以在不改变结果的情况下置换这些碎片。这种观点也出现在更抽象的设置中，例如用线性方程以平衡的方式分解数据集以有效地处理它，或者在固定的点集中评估函数以计算其积分。请注意，如果我们将参与方的主观偏好引入到公平分配问题中，固有的对称性就会消失。同样，这种对称性的丧失对于到目前为止所提到的这类问题来说也是一种自然现象。该项目开发了解决这些问题的方法，即使在缺乏对称性的情况下。值得注意的是，这些方法是拓扑的，也就是说，是连续的，而这些问题具有组合的，或离散的味道。因此，这项研究在数学的不同分支之间建立了桥梁，使一个分支中的问题适用于另一个分支的工具。PI将从几何观点出发，对一些组合问题进行研究。研究的主题包括集合的交集模式，包括在纯粹的组合环境（如超图匹配和设计理论）以及几何环境（如凸集的交集和复形到欧几里得空间的不可嵌入性），以及阿贝尔群中结构化集合的研究（如和集，包含零和的集合结构和球形设计）。这些问题，其中一个感兴趣的是构建一个对象或显示其不存在，但全球障碍出现-本地解决方案不粘在一起，形成所需的对象。几何学和拓扑学提供了一个合适的框架来组织这些非局部现象，并在大范围内检测这些障碍物。这项研究的重点是承认几何概括的问题，旨在从更灵活的几何对应物中划分刚性离散结果。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。

项目成果

Spaces of embeddings: Nonsingular bilinear maps, chirality, and their generalizations

嵌入空间：非奇异双线性映射、手性及其概括

• 发表时间: 2022
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Frick, Florian;Harrison, Michael
• 通讯作者: Harrison, Michael

Vietoris thickenings and complexes have isomorphic homotopy groups

• DOI: 10.1007/s41468-022-00106-5
• 发表时间: 2022-06
• 期刊: Journal of Applied and Computational Topology
• 作者: Henry Adams;F. Frick;Žiga Virk

The Topology of Projective Codes and the Distribution of Zeros of Odd Maps

射影码的拓扑与奇映射的零点分布

• DOI: 10.1307/mmj/20216170
• 发表时间: 2023
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Adams, Henry;Bush, Johnathan;Frick, Florian
• 通讯作者: Frick, Florian

Coupled embeddability

耦合嵌入性

• DOI: 10.1112/blms.12646
• 发表时间: 2022
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Frick, Florian;Harrison, Michael
• 通讯作者: Harrison, Michael

METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES

• DOI: 10.1112/mtk.12010
• 发表时间: 2019-07
• 期刊: Mathematika
• 影响因子: 0.8
• 作者: Henry Adams;Johnathan Bush;F. Frick