A Continuous Analogue to Sperner's Theorem

John Eveland

Friday, April 30, 2004
11:00 am, 845 Patterson Office Tower

Abstract:

Sperner's theorem, which is well-known in extremal combinatorics, asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient n choose floor{n/2}. It would be nice to find a similiar result on the infinite lattice Mod(n) of linear subspaces (through the origin) of the vector space R^n. It will be shown that there exists an invariant measure, up to a unique constant, on the elements of this lattice of rank k, that is, the set of all subspaces of dimension k of R^n. This is the Grassmannian. Using this measure, a continuous analogue to Sperner's theorem will be given. This talk will examine the paper, by Klain and Rota, of the same name.