  This is a blend of two informal reports on the activities of the seminar on
Galois representations and mirror symmetry given at the Conference on
classification problems and mirror duality at the Steklov Institute, in March
2006, and at the Seminar on Algebra, Geometry and Physics at MPI, in November
2007. We assess where we are on the issue of the spectra of Fano varieties, and
state problems. We introduce higher dimensional irreducible analogues of
dessins, the low ramified sheaves, and hypothesize that Fano spectra relate to
their geometric conductors. We give a recipe to a physicist.
