From data to cohomologies of digraphs, to representations of categories

Motivated by applications to topological data analysis, in the last years we have been witnessing a growing interest in new (topological, homological, categorical) features associated to graphs. The aim of the talk is to describe a recently developed (co)homology theory of directed graphs, called multipath cohomology, whose definition is inspired by Khovanov homology of knots. We will show its relations with more established invariants, such as chromatic homology of graphs, Hochschild homology of algebras, and matching complexes. If time allows it, we will show that the ranks of multipath cohomology, when restricted to graphs with bounded genus, grow at most polynomially in the number of edges, and that the order of its torsion is bounded. This will depend upon structural properties of (the representation category of) graphs, such as its Noetherianity properties.