The magnitude of metric spaces I


Venue   Advanced Course on Integral Geometry and Valuation Theory, Centre de Recerca Matemàtica, Barcelona, 7 September 2010.

Abstract   Magnitude is a new and powerful invariant of metric spaces. Its origins are in category theory, but it also seems to pull together, in a subtle way, many important invariants of integral geometry.

Magnitude is most readily defined for finite metric spaces. Although this is interesting in its own right (and has been used to solve a problem to do with the maximization of biodiversity), it is when one passes to infinite spaces that one sees the connections with other, well-known invariants. To see this, we first note that magnitude does not change predictably as a space is scaled. Hence, given a space X, one gains information about X by graphing the magnitude of tX (that is, X scaled up by a factor of t) against the positive real number t. For instance, in certain families of examples, one can extract from this graph of magnitudes both the dimension of X and all of its intrinsic volumes. In that sense, magnitude contains these other invariants.

This is the first of two talks. The second will be given by Simon Willerton.

Reference   Tom Leinster and Simon Willerton, On the asymptotic magnitude of subsets of Euclidean space (2009).

Slides   In this pdf file. The slides from part II, by Simon Willerton, are here.

Discussion   Integral geometry in Barcelona

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