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The many faces of magnitude

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Mathematics Colloquium

The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, produces a family of invariants in different contexts across mathematics. All of them can be loosely understood as “size”. For example, the magnitude of a convex set is an invariant from which one can conjecturally recover many important geometric measures: volume, surface area, perimeter, and so on. The magnitude of a graph is a new invariant that shares features with the Tutte polynomial but is not a specialization of it. The magnitude of a category is very closely related to the Euler characteristic of a topological space. Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. I will give an aerial view of this landscape.

This talk is part of the Mathematics Colloquium series.

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