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Graph theory in higher dimensions

高维图论

基本信息

批准号：
EP/V009044/1
负责人：
Agelos Georgakopoulos
金额：
$ 48.16万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV009044%2F1
关键词：
Graph theory higher dimensions

项目摘要

Graph theory is a modern and highly active branch of mathematics with an increasingly important impact on other areas, including computer science, geometry, number theory, topology, probability, and statistical mechanics. A mainstream trend in graph theory aims at generalising results from graphs to hypergraphs. Such gen- eralisations tend to be much harder than their graph analogues, or even provably impossible, and so despite the career-long efforts and deep machinery of generations of graph theorists, we will never be able to extend all of graph theory to hypergraphs. But it is definitely worth doing so for those statements likely to have an impact on other disciplines.This proposal takes this viewpoint as a starting point. It departs from the mainstream in two ways. Firstly, the generalisations we seek are driven by concrete applications to other areas of mathematics highlighted by the objectives set below. Secondly, rather than a purely combinatorial approach to graphs and hypergraphs, we take a topological viewpoint: when graphs are viewed as 1-dimensional simplicial complexes, natural topological extensions of definitions and theorems to higher-dimensional cell-complexes -the alter ego of hypergraphs- suggest themselves.One of the ambitions of this project is to advance the development of a unified theory of low-dimensional topological combinatorics that parallels the growth of graph theory into a discipline and has a long-lasting impact on other disciplines. To avoid the risk of getting lost in abstract theory building, concrete objectives are set out below to ensure that the theory grows into the right directions and bears fruit within the time-frame of the project. Planarity, and its higher-dimensional analogues, plays an important role throughout providing some common ground for the various objectives. These objectives are both important and timely, being strongly related to seminal recent advances.

图论是数学的一个现代和高度活跃的分支，对其他领域的影响越来越重要，包括计算机科学，几何，数论，拓扑，概率和统计力学。图论中的一个主流趋势是将结果从图推广到超图。这样的一般化往往比它们的图类似物要困难得多，甚至是不可能的，因此，尽管几代图论家在职业生涯中付出了长期的努力和深入的研究，我们永远无法将所有的图论扩展到超图。但对于那些可能对其他学科产生影响的言论来说，这样做绝对值得。本提案以这一观点为出发点。它在两个方面偏离了主流。首先，我们所寻求的概括是由以下目标所强调的其他数学领域的具体应用所驱动的。其次，我们采用拓扑观点，而不是纯粹的组合方法来研究图和超图：当图被视为1维单纯复形时，自然拓扑扩展的定义和定理，高维细胞复合物-超图的另一个自我-建议自己。这个项目的野心之一是推进低，维拓扑组合学，平行的图论发展成为一门学科，并对其他学科产生了长期的影响。为了避免在抽象的理论建设中迷失方向的风险，下面列出了具体的目标，以确保理论朝着正确的方向发展，并在项目的时间范围内取得成果。平面性及其高维类似物在为各种目标提供一些共同基础的过程中发挥着重要作用。这些目标既重要又及时，与最近的重大进展密切相关。