Quiver representations in 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.