Mapping Class Group of Surfaces and Geometry of Moduli Spaces

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In this project, we focussed on the study of the structure of the mapping class group of surfaces (m.c.g. for short) as well as the moduli space of compact Riemann surfaces, together with various problems closely related with this. They include the following thema : cohomology group of m.c.g., the theory of the Floer homotopy types, topological invariants based on gauge theory, construction of the harmonic Magnus expansion of m.c.g., structure of the Grothendieck-Teichm＼"uller group, the volume conjecture, non-commutative geometry in dimensions 3,4, finite subgroups of m.c.g., the Jones representation of m.c.g., relation between m.c.g. with 4-dimensional topology. From the interactions of these thema, we found new directions of research such as the relation between the geometry of m.c.g. and the symplectic topology as well as the comparaison between m.c.g. and the outer automorphism group of free groups.