Constructions of cycles in the moduli space of Riemann surfaces and the moduli space of graphs

In this talk, I would like to discuss various methods of constructing (co)cycles in the moduli space of Riemann surfaces and the moduli space of graphs. More precisely, I will mention three methods which make use of

1. the theory of group cohomology,
2. the natural cell decompositions of the moduli spaces and
3. the structure of the derivation algebra of surfaces via a theorem of Kontsevich.

A particular emphasis will be given on the comparison between the above two moduli spaces.