One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall the version of the theorem that I learned in an undergraduate class in model theory.

An algebraic map $F: \mathbb{C}^{n} \to \mathbb{C}^{n}$ is injective iff it is bijective.

The model theoretic proof of this result is very simple. Without discussing details ( truth in TACF_0 is the same as truth in TACF_p for p big enough )one can replace $\mathbb{C}$ by the algebraic closure of $F_p$ and prove the result there. Although the algebraic proof is also "simple" (Hilbert's Nullstellensatz is what one needs ) I personally think that the model theoretic proof is great by its simplicity and elegance.

So my first question is: Are there more examples like AX's theorem in which the model theoretic proof is much more simple that the ones given by other areas. (I should mention that some algebraist I know consider quantifier elimination for TACF non a model theoretic fact, so for them the proof that I refer above is an algebraic proof)

My second question: One of the most famous model theoretic applications to algebra and number theory is Hrushovski's proof on Mordell-Lang of the function field Mordell-Lang conjecture. I'd like to know what are the research questions that applied model theorists are currently working on, besides continuation to Hrushovski's work . In particular, I'd like to know if there is any model theorist that work in applications to Iwasawa theory.