I was attending a conference on geometric group theory, geometric analysis, and mapping class group in the Japan-U.S. Mathematics Institute (JAMI) of JHU at Baltimore last weekend. JAMI with a history of about twenty years holds conference every year for promoting communications and cooperations of American mathematicians and Japanese mathematical society.

Mapping class group , the group of orientation-preserving self-homeomorphisms of a topological space, becomes another interesting algebraic object in recent decades besides homology groups as algebraic invariants for topological spaces. Elements in mapping class group induce actions on homology groups of a space, the kernel called Torelli group is important for understanding mapping class group because of the following short exact sequence

    (1)

where is the symplectic group of degree over , and is a genus surface. There are many interesting investigations on and the so-called Johnson homomorphism that gives the free part of the abelianization of .