Magnitude cohomology of graphs


Venue   SIAM Conference on Applied Algebraic Geometry 2023, minisymposium on Applications of Magnitude and Magnitude Homology to Network Analysis, Eindhoven University of Technology, Netherlands, 10 July 2023

Abstract   While magnitude homology has received a great deal of recent attention, magnitude cohomology has yet to catch up. The theory of magnitude cohomology is due to Richard Hepworth (Mathematische Zeitschrift 301 (2022), 3617-3640). It is defined in the wide generality of enriched categories and specializes to metric spaces and, therefore, graphs. Like many cohomology theories, it has more structure than its homological counterpart. In particular, there is a cup-like product, which, however, may not be commutative if the enriching category is not cartesian.

Hepworth proves the very striking result that a metric space is completely determined up to isometry by its magnitude cohomology (seen as a graded ring), provided that the space is sufficiently discrete: there is a lower bound on the set of nonzero distances. Since graphs are discrete in this sense, magnitude cohomology defines an embedding of the class of graphs into the class of graded rings.

I will give an overview of the world of magnitude cohomology, concentrating on the case of graphs.

Slides   In this PDF file.

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