Variants of Mordell-Lang

I will report on two recent papers with D. Ghioca and U. Zannier (joined by P. Corvaja and F. Hu, respectively) in which we consider variants of the Mordell-Lang conjecture.  In the first of these, we study the dynamical Mordell-Lang conjecture in positive characteristic, proving some instances, but also showing that in general the problem is at least as hard as a difficult diophantine problem over the integers.  In the second paper, we study the Mordell-Lang problem for extensions of abelian varieties by the additive group.  Here we have positive results in the function field case obtained by using the socle theorem in the form offered as an aside in Hrushovski's 1996 paper and in the number field case we relate this problem to the Bombieri-Lang conjecture.