CAREER: Geometric and Topological Combinatorics

职业：几何和拓扑组合学

• 批准号: 2042428
• 负责人: Florian Frick
• 金额: $ 40万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2042428&HistoricalAwards=false
• 关键词: CAREER; Geometric; Topological; Combinatorics

Geometry and topology are mathematical branches that provide methods that measure phenomena that are global instead of local. If a problem's resolution depends on the aggregate of its information then geometric methods are useful in detecting global obstructions. Numerous problems across mathematics and its applications are global in this sense, ranging from data science to economics. The research supported by this grant will develop new topological and geometric methods to tackle problems further afield, primarily discrete, non-continuous problems. This promises new insights at the confluence of combinatorics, geometry, and topology. Students at all stages will be involved in the research effort supported by this grant.Among the general goals of the research program are the following: In applications of equivariant topology one often requires that a certain parameter has to be a prime power, and methods fail outside of this prime power case. Recent research of the PI has suggested that one may effectively circumvent this prime power requirement via a synthesis of topological and combinatorial techniques. This will be further developed. The application of topological methods brings about geometric generalizations of combinatorial problems. To understand the limitations of topological techniques one has to study those types of problems, where geometric results deviate considerably from their combinatorial special cases. A central goal will thus be to delimit rigid combinatorial results from their flexible geometric counterparts. In addition to showing the existence of a solution to some combinatorial problem, one is often interested in quantifying how rich the space of solutions is. General topological methods will be developed that provide lower bounds for the topology of the space of solutions of a given combinatorial problem.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.

几何学和拓扑学是数学分支，提供了测量全局而不是局部现象的方法。如果一个问题的解决方案取决于其信息的聚合，那么几何方法在检测全局障碍物时是有用的。从这个意义上说，数学及其应用中的许多问题都是全球性的，从数据科学到经济学。该资助支持的研究将开发新的拓扑和几何方法来解决更远的问题，主要是离散、非连续问题。这预示着在组合学、几何学和拓扑学的交汇处有新的见解。所有阶段的学生都将参与这项资助支持的研究工作。研究计划的总体目标如下：在等变拓扑的应用中，人们经常要求某个参数必须是素数幂，而方法在这个素数幂情况之外失败。PI最近的研究表明，人们可以有效地规避这一主要的权力要求，通过综合的拓扑和组合技术。这将得到进一步发展。拓扑方法的应用带来了组合问题的几何推广。要了解拓扑技术的局限性之一，必须研究这些类型的问题，其中几何结果大大偏离其组合的特殊情况。因此，一个中心目标将是从灵活的几何对应物中划分出刚性的组合结果。除了显示一些组合问题的解的存在性之外，人们通常对量化解的空间有多丰富感兴趣。一般的拓扑方法将被开发，提供一个给定的组合problem.This奖项反映了NSF的法定使命的解决方案的空间拓扑的下限，并已被认为是值得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的支持。

项目成果

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

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