Given a complex abelian variety, there is an easy compatibility between its group law and its  singular cohomology; Beauville (and later work of Deninger-Murre) explained how to extend this structure to study algebraic cycles on families of abelian varieties, using a Fourier-type transform.  I will survey this work and discuss to what extent it can be extended to degenerating families (also how it fits in with the first talk, if I have time)