Low dimensional linear representations of mapping class groups

The action of the mapping class group of a compact orientable surface of genus g on the first
homology group of the surface gives  the classical symplectic representation of dimension 2g.
Recently,  John Franks and Michael Handel proved that complex representations of dimension
$n \leq 2g-4$ of the mapping class groups are trivial. In this talk, I will show that all representations
of dimension $n \leq 2g-1$ are trivial, improving the result of Franks and Handel. I will also discuss t
he analogous problem for mapping class groups of nonorientable surfaces.