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.