A finite simplicial complex X is just a collection of subsets of a finite set (of vertices) V, where X is closed under taking subsets. Members of X are referred to asfaces or simplices, and this reflects the geometric side of the story. Indeed, simplicial complexes are fundamental in many parts of geometry and topology. A face A is said to have dimension |A|-1 and the dimension of X is the largest dimension of any face in X. In particular, a graph is just a one-dimensional simplicial complex. Several years ago Roy Meshulam and I defined a model of random simplicial complexes which in one dimension coincides with Erdos and Renyi's classical G(n,p) model. I will review some of the progress made in this area and tell some other surprising things discovered in these investigations.

My collaborators in this endeavor were: Roy Meshulam, Tomasz Luczak, Ilan Newman and Yuri Rabinovich. Much of the work was done by my students Lior Aronshtam and Yuval Peled.

 

The talk will be largely self-contained. I do not assume earlier familiarity with simplicial complexes.