  For a oriented genus g surface with one boundary component, S, the Torelli
group is the group of orientation preserving homeomorphisms of S that induce
the identity on homology. The Magnus representation of the Torelli group
represents the action on F/F" where F=pi_1(S) and F" is the second term of the
derived series. We show that the kernel of the Magnus representation, Mag(S),
is highly non-trivial and has a rich structure as a group. Specifically, we
define an infinite filtration of Mag(S) by subgroups, called the higher order
Magnus subgroups, M_k(S). We develop methods for generating nontrivial mapping
classes in M_k(S) for all k and g>1. We show that for each k the quotient
M_k(S)/M_{k+1}(S) contains a subgroup isomorphic to a lower central series
quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally We show that for g>2 the
quotient M_k(S)/M_{k+1}(S) surjects onto an infinite rank torsion free abelian
group. To do this, we define a Johnson-type homomorphism on each higher order
Magnus subgroup quotient and show it has a highly non-trivial image.
