Pure Mathematics Colloquium,
Queen Mary, University of London,
6 February 2012.
Abstract Many invariants of "size" in mathematics are tied together by a single invariant, the Euler characteristic of a category. This phenomenon can be found in contexts from orbifolds to associative algebras, taking in such concrete invariants as volume and (conjecturally) perimeter of convex sets. Closely related are invariants of "spread", typified by information entropy. For example, the original category-theoretic invariant turns out to solve a problem in mathematical ecology: that of how (theoretically) to maximize biological diversity. I will conduct a tour through all of this, assuming no knowledge of anything in particular.
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