Magnitude and magnitude homology


Venue   University of Aberdeen, 14 February 2018

Abstract   Magnitude is an invariant defined in the very wide generality of enriched categories, which when interpreted in particular types of enriched category gives familiar invariants such as cardinality and Euler characteristic. But it is when interpreted for metric spaces that it gives something really new: a real invariant of metric spaces that simultaneously encodes classical invariants such as volume, curvature and dimension, and more besides. Moreover, magnitude applied to finite metric spaces appears to pick up salient features of point clouds. I will show some empirical evidence.

A few years ago, Hepworth and Willerton demonstrated that in the special case of graphs (seen as metric spaces with integer distances), magnitude is the Euler characteristic of a certain graded homology theory, "magnitude homology". This is similar in spirit to Khovanov's realization of the Jones polynomial as the Euler characteristic of Khovanov homology. Recent work by Shulman has generalized magnitude homology to a large class of enriched categories - including metric spaces. And the magnitude homology of a metric space captures even more geometric information than magnitude itself. For instance, the vanishing of first magnitude homology is equivalent to convexity.

I will give an overview of all this, starting from scratch.

Slides   In this pdf file. See this bibliography for references and further reading.

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