The peak algebra, introduced by John Stembridge in connection with the combinatorial study of shifted tableux and Schur Q-functions, has recently emerged as a natural algebraic setting to study flag enumeration in Eulerian posets, in particular face-lattices of convex polytopes.
I will discuss the peak algebra and Eulerian enumeration in the broader context of the following peak phenomenon:
Given a statement regarding the algebra of quasisymmetric functions (these include the symmetric functions), there is an analagous statement that holds for the peak subalgebra (these include Schur's Q-functions).
I will give some examples of this phenomenon, and explain what implications it might have for the study of flag f-vectors, the cd-index, and related invariants on posets.