Quiver representations in Topological Data Analysis

拓扑数据分析（TDA）是使用代数拓扑中的抽象工具来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，这些有限点云携带的拓扑信息很少。为了在不同尺度上研究它们，我们分配了过滤的简单复合体，即一个有限增长的简单复合体序列，其顶点与给定点重合。这通常是通过使用切赫或Vietoris-Rips复合物来实现的，如b[1]中所述。使用这个简单复形族，我们可以计算一个域上过滤复形的持久同调，从而获得给定数据集在一个不断增加的尺度族上的拓扑洞察力。在简单型具有不同重要性级别的应用程序中，将代数权重与简单型关联起来可能是有用的。这样做的一种可能性是使用从单纯复形的简单偏序集到域[2]上的向量空间范畴的函子。这种方法产生了新的上同调理论，并对理论和计算提出了新的挑战。箭矢是具有多条边和自环的有向图。给定箭袋的表示为箭袋的每个顶点分配一个向量空间，并为其每个箭头分配一个线性映射。颤振表示通常因其可分解性而被研究，但它们也是简单复合体上的束概念的推广。这种将轴视为颤振表示的方法设计了一种有效的算法来计算简单复合体的第0轴上同调。这个项目的目标是进一步发展上面描述的方法，在这个方法中，滑轮被看作是颤抖的表现。我们希望这种捆和颤振表示之间的类比将在理论和计算上为拓扑数据分析带来有趣的发展。两个主要目标将是寻找这种方法的可能应用，并为高阶拓扑不变量创建新的算法。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的“几何和拓扑”研究领域。

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.