In this talk we will explore the topological features of random simplicial complexes. We will introduce two models that extend the Erdős–Rényi model for generating random graphs. Unlike in the one-dimensional case (random graphs), we may pose several interesting topological questions. For example, the fundamental groups of random complexes can be seen as a model for generating random groups, which may be of independent interest. On the other hand, random simplicial complexes arise naturally in modelling the real world; eg shape recognition, data analysis, configuration spaces of physical systems, etc.  

We will focus on random 2-dimensional simplicial complexes and discuss their geometric and topological properties under different probability measures.  One question of interest is whether one can generate randomly aspherical 2-complexes (i.e. such that \pi_2(Y) = 0) and whether random aspherical 2-complexes satisfy the Whitehead Conjecture.  This conjecture, proposed in 1941 by J.H.C. Whitehead, states that any subcomplex of a aspherical 2-complex is also aspherical. If time allows we will show that, under a variety of natural probability measures, random aspherical 2-complexes satisfy the Whitehead Conjecture, almost surely. 

The talk is based on a joint work with Michael Farber.