Beijing Institute of Mathematical Sciences and Applications

This course is devoted to the notion of magnitude homology and its relation to Euler characteristic of a finite category and GLMY-theory for digraphs. Tom Leinster in 2006 introduced a notion of Euler characteristic of a finite category, which is a rational number (not necessarily integer) that coincides with the Euler characteristic of the classifying space, when it is defined. Later he generalized this notion to enriched categories and specified it to metric spaces, treated as enriched categories over the category of real numbers. This new invariant of a metric space is called the magnitude. Later Hepworth, Willerton discovered that the magnitude of a graph, treated as a metric space, can be presented as the alternating sum of dimensions of some version of homology of a graph, that they called magnitude homology. Finally Asao generalized this theory to the case of directed graphs and proved that for a directed graph there is a spectral sequence converging to the homology of the preorder defined by the digraph, whose first page is the magnitude homology and the main diagonal of the second page is the path homology introduced by Grigor’yan, Lin, Muranov, Yau. All these relations will be discussed in our lectures.