Abstract:

The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) originated as a correlation inequality in statistical mechanics. It has many applications in discrete probability and extremal combinatorics.

In this talk we derive a polynomial coefficient-wise inequality that refines the original FKG inequality. This polynomial FKG inequality has applications to $f$-vectors of joins of simplicial complexes, to Betti numbers of intersection of certain Schubert varieties, and to power series weighted by Young tableaux. The latter case includes a correlation inequality for the so called poissonization of Plancherel measure on symmetric groups, a probability measure on the set of all integer partitions.

The talk will be quite elementary and no previous familiarity with these topics will be assumed.