On Maeda's Conjecture

In 1997 Maeda established a conjecture in his work with Hida
that has been popularized in the following form:
the characteristic polynomials of the Hecke operators  acting on the
space of cusp forms of fixed weight for SL(2,Z) are all irreducible and
have full Galois group. Beyond Maeda-Hida's work, this conjecture, if
true, would have different consequences; for example for the
non-vanishing of L-functions or the inverse Galois problem. We will
explain the significance of the conjecture and discuss some recent
progress.