by the work of griffiths, the cohomology of a family of complex projective manifolds determines a period map from the base of the family to the quotient of a flag domain d. in the case where d is hermitian symmetric, these quotients admit a number of partial compactifications including the baily-borel and toroidal amrt compactifications. i describe recent work with matt kerr on computing the mumford-tate group of the analogs of the armt boundary components of a degeneration of hodge structure arbitrary weight.