Quiver representations in Topological Data Analysis

拓扑数据分析中的 Quiver 表示

• 批准号: 2580665
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 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=studentship-2580665
• 关键词: Quiver; representations; Topological; Data; Analysis

Topological Data Analysis (TDA) is the use of abstract tools from algebraic topology for the concrete task of analysing large and complex datasets. One of the more common data sets TDA applies well to do are finite point clouds in metric spaces. At first glance, as discrete spaces, these finite point clouds carry little topological information. In order to study them across various scales, one assigns filtered simplicial complexes, that is a finite increasing sequence of simplicial complexes, whose vertices coincide with the given points. This is generally achieved using the Cech or Vietoris-Rips complex, as explained in [1]. Using this family of simplicial complexes, we may compute the persistent homology of the filtered complex over a field to obtain a topological insight into our given data set across an increasing family of scales. In applications where simplices have different levels of importance, it may be useful to associate algebraic weights to the simplices. A possibility to do this is to use sheaves which are functors from the poset of simplices of a simplicial complex to the category of vector spaces over a field [2]. This approach produces new cohomology theories with new theoretical and computational challenges.A quiver is a directed graph with multiple edges and self-loops. A representation of a given quiver assigns a vector space to each vertex of the quiver and a linear map to each of its arrows. Quiver representations are often studied for their decomposability [3] but they are also a generalisation of the notion of sheaves on a simplicial complex. This approach of seeing sheaves as quiver representations led to the design of an efficient algorithm to compute the 0th sheaf cohomology of a simplicial complex. The goal of this project is to further develop the approach described above where sheaves are viewed as quiver representations. We hope that this analogy between sheaves and quiver representations will produce interesting developments for topological data analysis both theoretically and computationally. The two main objectives will be finding possible applications of this method and creating new algorithms for higher order topological invariants. It will require tools from diverse fields of mathematics including algebraic topology, representation theory, category theory and graph theory. This project falls within the EPSRC 'Geometry and Topology' research area.

拓扑数据分析（TDA）是使用代数拓扑的抽象工具来执行分析大型复杂数据集的具体任务。 TDA 最常用的数据集之一是度量空间中的有限点云。乍一看，作为离散空间，这些有限点云携带很少的拓扑信息。为了在不同尺度上研究它们，我们分配了过滤单纯复形，即单纯复形的有限递增序列，其顶点与给定点重合。这通常是使用 Cech 或 Vietoris-Rips 复合体来实现的，如 [1] 中所述。使用这个单纯复形族，我们可以计算过滤后的复形在域上的持久同源性，以获得对不断增加的尺度族中给定数据集的拓扑洞察。在单纯形具有不同重要性级别的应用中，将代数权重与单纯形相关联可能很有用。实现此目的的一种可能性是使用滑轮，滑轮是从单纯复形的单纯形偏序集到域上向量空间类别的函子 [2]。这种方法产生了新的上同调理论，带来了新的理论和计算挑战。箭袋是具有多条边和自环的有向图。给定箭袋的表示将向量空间分配给箭袋的每个顶点，并将线性映射分配给其每个箭头。人们经常研究箭袋表示的可分解性 [3]，但它们也是单纯复形上滑轮概念的推广。这种将滑轮视为箭袋表示的方法导致设计了一种有效的算法来计算单纯复形的第 0 个滑轮上同调。该项目的目标是进一步开发上述方法，其中滑轮被视为箭袋表示。我们希望滑轮和箭袋表示之间的这种类比将为拓扑数据分析在理论和计算上产生有趣的发展。两个主要目标是寻找该方法的可能应用以及为高阶拓扑不变量创建新算法。它将需要来自不同数学领域的工具，包括代数拓扑、表示论、范畴论和图论。该项目属于 EPSRC“几何和拓扑”研究领域。