Abstract: I will review the Fourier-Mukai transform and some applications. This an equivalence of derived categories of coherent sheaves on complex tori (or complex torus fibrations), analogous to the classical Fourier Transform. I will give two examples of how this equivalence persists in families. Such families include non-commutative deformations, gerbey deformations, and families of fibrations over the same base carrying gerbes. If time permits, I will explain how this could relate to a general statement of Polishchuk and Rothstein.

The results discussed are partially in collaboration with J. Block, and T. Pantev