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Magnitude of Metric Spaces

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If you have a question about this talk, please contact Paul Taylor.

The notion of Euler characteristic is found all across mathematics, with the notions of dimension and cardinality being special cases. Notions of Euler characteristic exist for finite sets, finite groups and finite posets. Tom Leinster generalized these three to a notion of Euler characteristic for finite categories. Then he generalized again to finite enriched categories. Metric spaces can be viewed as a type of enriched category and so we obtained a notion of Euler characteristic which Tom and I named the magnitude of a metric space. This can be thought of an ‘effective number of points’ of the metric space. I will explain all of the above in the first part of the talk.

Following calculations, it became clear that the notion of magnitude could be extended to infinite metric spaces such as compact subsets of a Euclidean space; however, it seems that it is hard to calculate the magnitude in general. In the second part of the talk I will explain how some interesting combinatorics of counting lattice paths enters the story for calculating the magnitude of balls in odd dimensional Euclidean spaces.

No previous knowledge of magnitude will be assumed.

This talk is part of the Theoretical computer science seminar series.

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