Entropy, diversity and magnitude

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.