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. 



Model theoretic applications to algebra and number theory(Iwasawa Theory)
It's hard for me to think of an area of algebra that applied model theorists **haven't** touched recently. I have not heard of any logicians working on Iwasawa theory, but it wouldn't surprise me if there are some.


Diophantine geometry: [here is a survey article by Thomas Scanlon](http://math.berkeley.edu/~scanlon/papers/bsl4ap00.pdf) on applications of model theory to geometry, including discussions of Mordell-Lang and the postivie-characteristic Manin-Mumford conjecture.


Number fields: [Bjorn Poonen has shown](http://www-math.mit.edu/~poonen/papers/uniform.pdf) that there is a first-order sentence in the language of rings which is true in all finitely-generated fields of characteristic 0 but false in all fields of positive characteristic. It was conjectured by Pop that any two nonisomorphic finitely-generated fields have different first-order theories.


Polynomial dynamics: [see here](http://math.berkeley.edu/~scanlon/papers/pd.pdf) for a recent preprint by Scanlon and Alice Medvedev. It turns out that first-order theories of algebraically closed difference fields where the automorphism is "generic" are quite nice.


Differential algebra: By some abstract model-theoretic nonsense ("uniqueness of prime models in omega-stable theories"), it follows that any differential field has a "differential closure" (in analogy to algebraic closure) which is unique up to isomorphism over the base field. There are much more advanced applications, e.g. [here](http://math.berkeley.edu/~scanlon/papers/jetB.pdf).


Geometric group theory: Zlil Sela has recently shown that any two finitely-generated nonabelian free groups are elementarily equivalent (i.e. they have the same first-order theory). According to [the wikipedia article,](http://en.wikipedia.org/wiki/Zlil_Sela) this work is related to his solution of the isomorphism problem for torsion-free hyperbolic groups, but I don't understand this enough to say whether this counts as an "application" of model theory.


Exponential fields: Boris Zilber has suggested a model-theoretic approach to attacking Schanuel's Conjecture. His conjecture that the complex numbers form a "pseudo-exponential field" is actually a *strengthening* of Schanuel's Conjecture, but the picture that it suggests is appealing. See [here](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture) for more.


This is in addition to the work on Tannakian formalism, valued fields, and motivic integration that have already been mentioned in other answers, and I haven't even gotten to all the work by the model theorists studying o-minimality. This was just a pseudo-random list I've come up with spontaneously, and no offense is meant to the areas of applied model theory that I've left off of here!


