ErdÅ‘s and RÃ©nyi initiated the study of the random graph

G(n,p), where n is the number of vertices and where each edge has

probability p. A celebrated theorem is that p = log n / n is a sharp

threshold for connectivity. In 2006 Linial and Meshulam defined

2-dimensional random simplicial complexes Y(n,p) and exhibited a

homological analogue of the Erdos-Renyi theorem.

Babson, Hoffman, and I studied fundamental group of these complexes

and found a very different story. To find a sharp exponent for the

vanishing threshold we combined techniques from combinatorial homotopy

theory and geometric group theory. Time permitting, I will discuss

more recent work showing that these random complexes satisfy certain

“expansion” properties, making them higher-dimensional analogues of

expander graphs.