Abstract: The Brauer Project (joint with Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nick Ham and James East) aims to classify, understand and enumerate the idempotents in several families of diagram semigroups and their algebras. We provide combinatorial and diagrammatic conditions for idempotency in certain of these, such as the partition monoid, the Brauer monoid, the partial Brauer monoid, their planar variations, and a novel family of semigroups which we call the partial Jones monoids.
The talk will focus on the characterisation and the recursive approach we can use to count in the non-planar cases, and, time permitting, I'll describe a cell complex which arises naturally in the planar cases whose structure informs much of what we know about the combinatorics and the structure of the ideals in these semigroups.