Fraïssé theory is a method in
classical Model Theory of producing canonical limits of certain families of
finite structures. For example, the random graph is the Fraïssé limit of the family of all finite graphs. It turns out that this method can be
dualized, with the dualization producing projective Fraïssé limits, and
applied to the study of compact metric spaces. I will describe recent results,
due to several people, on connections between projective Fraïssé limits and the structure of some canonical compact spaces and their
homeomorphism groups (the pseudoarc, the Menger curve, the Lelek fan, simplexes
with the goal of developing a projective Fraïssé homology theory).