We will discuss in this talk two different ways of building random simplicial complexes. These constructions give us workable notions of random topological spaces, and we can ask questions about their typical properties as we vary parameters. For example, is a typical space likely to be simply connected? Or, how large should we expect the rank of the kth homology group to be? Answers can be made fairly precise, and it turns out that there are sharp phase transitions between these algebraic invariants being nontrivial and trivial. In our results, we find analogues of Gromov's phase transition for random groups (from hyberbolic to trivial), as well as generalizations of the Erdos-Renyi theorem for connectivity of the random graph to higher dimensions. The probability methods are elementary, and the talk will aim to be self contained. Part of this is joint work with Eric Babson and Chris Hoffman. |