A recurring task in data science is understanding the connectivity of information. Agebraic topology offers us tools for understanding, quantifying and comparing topological features of data clouds. Although, determining the existence homeomorphism is out of reach for general computation, compromise in determining its non-existence is offered trough extension of Euler Number Invariant – the notion of homology, and more percisely, the notion of persistent homology: the homological information conserved over different scales of some parameter, usually represented in form of a barcode – a diagram representative of vanishing and persistence of homology groups.

Persistent Homology has proven itself as a powerful tool in analyzing the topology of data point clouds. Notion of persistence now yields numerous applications in fields of Computer Vision, Data Clustering, Sensor Coverage, Neuroscience, Genomics and even Thermodynamics. This paper discusses an approach to representation of persistent homology via quivers –  directed graphs for which it is permitted to have multiple arcs, which, when interpreted algebraically, in a representation-theoretic view, yield results in theory of multidimensional persistence, providing us with ability to consider more than one scale parameter to analyze a data set, providing us with a way to see the fuller picture of how our data cloud is interconnected.

This topic is dear to me as it touches on theoretical computer/data science as well as algebraic topology, the two topics I harbor a great interest in. Although often disjoint, they are “married” in TDA. Often, (well not very often, but if it happens its a big problem), when working with data analysis and machine learning one might run into a problem of the most efficient way to separate/cluster the data that happens to be arranged in a topologically non-trivial way. TDA is able to provide great help in such cases, as well as yield new ways to visualize and understand relations of entries within the point cloud.