  The evolution of the galaxy stellar mass--star formation rate relationship
(M*-SFR) provides key constraints on the stellar mass assembly histories of
galaxies. For star-forming galaxies, M*-SFR is observed to be fairly tight with
a slope close to unity from z~0-2. Simulations of galaxy formation reproduce
these trends owing to the generic dominance of smooth and steady cold accretion
in these systems. In contrast, the amplitude of the M*-SFR relation evolves
markedly differently than in models. Stated in terms of a star formation
activity parameter alpha=(M*/SFR)/(t_H-1 Gyr), models predict a constant
alpha~1 out to redshifts z=4+, while the observed M*-SFR relation indicates
that alpha increases by X3 from z~2 until today. The low alpha at high-z not
only conflicts with models, but is also difficult to reconcile with other
observations of high-z galaxies. Systematic biases could significantly affect
measurements of M* and SFR, but detailed considerations suggest that none are
obvious candidates to reconcile the discrepancy. A speculative solution is
considered in which the stellar initial mass function (IMF) evolves towards
more high-mass star formation at earlier epochs. Following Larson, a model is
investigated in which the characteristic mass Mhat where the IMF turns over
increases with redshift. The observed and predicted M*-SFR evolution may be
brought into agreement if Mhat=0.5(1+z)^2 Mo out to z~2. Such evolution broadly
matches recent observations of cosmic stellar mass growth, and the resulting
z=0 cumulative IMF is similar to the paunchy IMF favored by Fardal et al to
reconcile the observed cosmic star formation history with present-day fossil
light measures. [abridged]
