Random simplicial complexes are one generalization of random graphs. In this seminar we will consider the model of 'permutations complexes', a simplicial complex obtained from a random permutations. More precisely, let \pi be a permutation of length n. Our model is the order complex of the partial order induced by \pi on the set [n].

We will discuss various topological properties of this complex, including homological dimension and homotopy type. Our main result is that w.h.p. the complex is r-connected for r=log(n)/loglog(n) . We will also show that the complex contains an octahedral sphere of dimension c\sqrt(n), and give an upper bound and a new lower bound on c.