I will discuss the relationship between the classical moduli space of Riemann surfaces and the configuration space of surfaces smoothly embedded in high dimensional euclidian space. The homological structure of the latter space can be completely determined when the genus of the embedded surfaces tend to infinity.
The configuration space may also be defined for manifolds of dimension larger than two. A recent theorem of Galatius and Randal-Williams identifies the homology of the configuration space of certain 2d-dimensional manifolds embedded in high dimensional euclidian space, the so called generalized surfaces. The above results concerning homology of moduli spaces do not give any information about the homotopy groups, mainly because the spaces are not simply connected. But for manifolds of dimension at least five,there is an alternative approach to the moduli spaces.
I will end the lecture with a brief description of joint work with Alexander Berglund concerning the homotopical and homological structure of the homotopy automorphism group of the generalized surfaces in terms of derivations of free Lie algebras and outer automorphisms of free groups. This represents the first step in the alternative approach.
The lecture will be as non technical as I can master.