The talk will be a survey of various geometric applications of the Fourier-Mukai transform via the notion of M-regularity for coherent sheaves.
I will first explain how most of (and in the case of abelian varieties essentially all of) the basic problems related to the embeddings and the equations of algebraic varieties can be formulated equivalently in terms of the global generation of appropriately chosen coherent sheaves.
I will then show how on (varieties mapping to) abelian varieties, the Fourier-Mukai transform allows one to obtain global generation from cohomological information, providing an analogue for Mumford's classical regularity for sheaves on projective space. This will be exemplified in lots of applications to the general study of abelian varieties, and also that of adjoint linear series on irregular varieties, moduli spaces of vector bundles, cohomology of symmetric products and Schottky-type statements.