Path Homology of Directed Graphs -
Homology theories for graphs are combinatorial invariants that can be used to compare and contrast graphs, and to try to understand how graphs are built out of smaller subgraphs. One particular flavor of graph homology is path homology, as defined by Grigor’yan, Lin, Muranov, and Yau. This invariant of directed graphs has a number of nice properties, which are analogous to properties of singular homology for topological spaces. This connection suggests that path homology is measuring something fundamental about graphs, although it is challenging to say exactly what. In this talk, I will introduce path homology in an accessible way: we will see the precise definition, but we will also do lots of concrete examples. As time allows, I will discuss an attempt, joint with Carranza, Doherty, Kapulkin, Sarazola, and Wong, to understand what type of information is encoded by path homology.