There is a nice characterization of forcing axioms in terms of generic ultrapowers with critical point omega-two (I believe this is due to Woodin). I will present this characterization and mention 2 applications:

1) Viale and Weiss' proof that PFA implies a strengthening of the Tree Property at omega-two (which resembles and strengthens a result of Krueger about separating the class of internally approachable structures from the internally club structures).

2) My results about diagonal stationary set reflection under MM and ``plus'' verions of MA(sigma-closed); these strengthen results of Foreman.

Viewing PFA and MM in this way (that is in terms of the existence of certain ideals on P_{omega-two)(theta) ) also suggests natural ways to strengthen the forcing axiom in question. For example, you can strengthen PFA by requiring that the relevant ideals satisfy certain properties like precipitousness, properness (of the forcing with the positive sets), decisiveness, etc.