Stanley's conjecture, cover depth and extremal simplicial complexes

A famous conjecture by R. Stanley relates the depth of a module, an algebraic invariant, with the so-called Stanley depth, a geometric one. We describe two related geometric notions, the cover depth and the greedy depth, and we study their relations with the Stanley depth for Stanley-Reisner rings of simplicial complexes. This leads to a quest for the existence of extremely non-partitionable simplicial complexes. We include several open problems and questions.This paper is a report about a research project suggested by J. Herzog at the summer school P.R.A.G.MAT.I.C. 2008 at the University of Catania. In particular, the paper describes a direction where we expect that possible counterexamples can be found at least for a weaker version of Stanley’s conjecture.