Magnitude meets persistence: homology theories for filtered simplicial sets

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the “effective number of points” of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. In their paper Leinster and Shulman list a series of open questions, two of which are as follows:

1) Magnitude homology only “notices” whether the triangle inequality is a strict equality or not. Is there a “blurred” version that notices “approximate equalities”?
2) Almost everyone who encounters both magnitude homology and persistent homology feels that there should be some relationship between them. What is it?

In this talk I will introduce magnitude and magnitude homology, answer these two questions and show that they are intertwined: it is the blurred version of magnitude homology that is related to persistent homology. Leinster and Shulman's paper can be found at arxiv.org/abs/1711.00802.