Combinatorics Seminar

In recent decades, partially ordered sets have been a fertile source of interesting discrete dynamics. The initial focus of researchers like Cameron, Fon-Der-Flaass, Panyushev, Armstrong, Stump, Thomas, Striker, and Williams was on combinatorics, but over time it has emerged that the combinatorial viewpoint is merely the tip (or rather the 0-skeleton) of an iceberg, and that the full story involves continuous piecewise-linear actions on polytopes. Furthermore, although the original emphasis was on the surprising way these maps tend to be periodic, my work with Einstein, Joseph and Roby has brought out another set of surprises: for most of these dynamical systems, many functions f have the property that f has the same average value on every orbit. This "homomesy phenomenon" is quite robust, and crops up even in cases where the orbit structure is exceedingly complicated.

I will give an accessible overview of this work requiring no knowledge of partially ordered sets beyond the basic notions of order ideals and antichains, and no knowledge of dynamical systems theory at all.

For those already familiar with work in this area, I will also discuss two recent results. One is a homomesy result for a nonperiodic action on a polytope, poised halfway between combinatorics and ergodic theory. The other is a viewpoint that uses linear algebra to build a bridge between homomesies for order ideal statistics and homomesies for antichain statistics. This linearization idea also works for dynamical systems associated with cluster algebras, though we do not know if it provides genuinely new information. Speaker(s): James Propp (University of Massachusetts, Lowell)