Noncrossing and nonnesting in finite Coxeter groups

There are beautiful algebraic generalizations of the poset of nonnesting partitions and the lattice of noncrossing partitions to all (crystallographic, at least) finite Coxeter groups. Both are counted by a generalized Coxeter-Catalan number and they share very refined enumerative features, yet it has been an open problem to provide any uniform bijection between these objects. In this talk I will explain the depth of this mystery and describe a conjectured uniform bijection. This is joint work with Hugh Thomas.