Quiver representations in Topological Data Analysis

拓扑数据分析(TDA)是使用代数拓扑学的抽象工具来分析大数据集和复杂数据集的具体任务。TDA最常用的数据集之一就是度量空间中的有限点云。乍一看，作为离散空间，这些有限的点云携带的拓扑信息很少。为了在不同的尺度上研究它们，人们指定过滤的单纯复形，这是一个有限递增的单纯复形序列，其顶点与给定点重合。这通常是使用切赫或越南里普斯复合体来实现的，如[1]中所解释的。使用这族单纯复形，我们可以计算域上过滤复形的持久同调，以获得对给定数据集在不断增加的标度族上的拓扑洞察。在简化具有不同重要程度的应用中，将代数权重关联到简化可能是有用的。这样做的一种可能性是使用层，它是从单纯复的偏序集到域上的向量空间范畴的函子[2]。这种方法产生了新的上同调理论，给理论和计算带来了新的挑战。箭图是具有多条边和自环的有向图。给定箭图的表示为箭图的每个顶点指定一个向量空间，为箭图的每个箭头指定一个线性贴图。箭图表示因其可分解性而经常被研究[3]，但它们也是单纯复形上层的概念的推广。这种将层看作箭图表示的方法导致了一个计算单纯复形的第0层上同调的有效算法的设计。该项目的目标是进一步发展上述方法，在该方法中，皮带轮被视为箭袋表示。我们希望，在理论上和计算上，这种层状表示和箭图表示之间的类比将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和创建高阶拓扑不变量的新算法。它将需要来自不同数学领域的工具，包括代数拓扑学、表示论、范畴理论和图论。本项目属于EPSRC‘几何与拓扑学’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.