Restrictions of Ultrapower Embeddings from Generic Extensions to their Ground Model

Assume that V[G] is a generic (set) forcing over the ground model V, and let \kappa be a measurable cardinal in V[G].

Let  j_W \colon V[G] \to M  be an ultrapower embedding associated with a normal measure W\in V[G]  on \kappa. 

What can be said about j_W\restriction_{V}? Is it definable in V? Is it an iterated ultrapower of V by its measures/extenders?

We will survey some known results. These results do not apply to the case where the forcing being used is an iteration of Prikry forcings. 

Our main focus will be to present new tools for dealing with iterations of Prikry forcings, taken with various supports (Nonstationary support, Full support and Easton support).

This is a joint work with Moti Gitik.