Abstract: 
The partition function on S^3 of supersymmetric field theories can be
reduced via localization to a matrix model. Equivalent theories under
some duality, like mirror symmetry, should have the same partition
function. This was indeed verified in many examples, usually by
employing increasingly complex integral identities. In my talk I will
point out that for certain theories which admit a free Fermi-gas
description (as I will review), mirror symmetry is simply a canonical
transformation on the phase space of the fermions.