Menger continua via projective Fraïssé theory

Projective Fraïssé theory was introduced by T.Irwin and S.Solecki as a natural framework for analyzing the dynamics of homeomorphism groups of compact metrizable spaces in terms of finite combinatorics. In this talk I will provide some basic background in projective Fraïssé theory and introduce a projective Fraïssé presentation of the universal Menger curve. I will then explain how various homogeneity and universality properties of this space reduce to standard Fraïssé theory and basic combinatorics. I will finally discuss extensions of this work to the Sierpinski carpet as well as to higher dimensions, where projective Fraïssé theory suggests the existence of homology versions of the n-dimensional Menger spaces and homology version the Hilbert cube.
Part of this is joint work with S. Solecki and part is joint work with T. Li, Z. Liu, V.V. Vakkada, and F. Weilacher.