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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.

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read this book"Model Theory:an introduction",by David Marker. –  Mr.Dolphin Nov 28 '09 at 8:58
    
To be fair, Marker's book is probably a good place to go if you're really serious about learning applied model theory. Though I've never read it myself, several people have told me that it's good, and it definitely has more of an "applied" bent than other textbooks on model theory. –  John Goodrick Dec 5 '09 at 16:29
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In many senses, the algebraist you mention is correct: QE for many theories is useful, but not quite a model theoretic fact (at least not in the same way as many of the other deeper model theoretic facts used in the constructions mentioned by Goodrick and others below). The details of QE proofs draw completely on the algebraic situation you're dealing with, and (mostly) depend on quite basic, quite soft model theory. Now, the examples mentioned by Medvedev (alg. dynamics), Goodrick and others usually depend on geometric model theory, not reducible to algebra, really. –  Andrés Villaveces Mar 16 '12 at 4:47
    
@Andrés: It's great to see your comment here-- when I wrote in the question "version of theorem that I learned in an undergraduate class in model theory" that was your class! –  Guillermo Mantilla Mar 17 '12 at 19:17

7 Answers 7

up vote 28 down vote accepted

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 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 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 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.

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, 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 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!

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I really like the theorem of Poonen and Scanlon you mentioned, but I could not find the paper in which they prove it. Do you have any reference where I can find it? I did not see it in their webpages –  Guillermo Mantilla Dec 2 '09 at 18:12
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@GMS: the proof was announced in a paper by Scanlon, "Infinite finitely-generated fields are biinterpretable with N," JAMS 21 (2008) 893-908, which is linked from his webpage. Unfortunately, it looks like he has recently retracted the proof. From his webpage: "There is a serious mistake in the proof of the definability of valuations. As it stands, Pop's Conjecture is still open." –  John Goodrick Dec 3 '09 at 3:56
    
...and I've edited my answer so that it links to a related (hopefully correct!) result of Poonen instead of Scanlon's slides. –  John Goodrick Dec 3 '09 at 3:58

I'd just like to expand on John Goodrick's mention of Zilber's work on exponential fields and mention that `Categoricity' of is an active area of research. In particular, model theory may be used to give some justification as to why theorems of classical mathematics should hold.

In general it's an interesting question to see what good model theoretic behaviour translates to in the world of classical mathematics.

One way of viewing things (and this is Zilber's point of view) is that if a mathematical structure is useful, and therfore well studied by the mathematical community, then it will be complicated enough to be interesting, but nice enough to be analysed. One aspect of model theory is involved with trying to classify structures with respect to how nice (or wild) they are (e.g. a structure could be strongly minimal, O-minimal, stable, categorical etc...).

At the top of the logical hierarchy sit categorical theories. A theory is $\kappa$-categorical if it has one model up to isomorphism in cardinality $\kappa$. The stereotypical example of a categorical theory is the theory of algebraically closed fields of characteristic $0$. The unique model of cardinality continuum is $\langle \mathbb{C}, + , \cdot , 0,1 \rangle$. This mathematical structure has pretty much every nice model theoretic property that you'd want in a structure - it's strongly minimal (definable sets are very simple i.e. either finite or cofinite), $\omega$-stable (there aren't many types of elements around), homogeneous (you can extend partial automorphisms to automorphisms of the whole structure), saturated (you can realise types - i.e. solutions to polynomials are in there). This theory is also complete and `categorical in powers' i.e. $\kappa$-categorical for every uncountable cardinal.

An amazing theorem of Morley actually says that if a first order theory is $\kappa$-categorical for one uncountable cardinal, then it is categorical for every uncountable cardinal. Morley's theorem (1965) kick started stability theory, and from there Shelah has developed an unbelievable amount of abstract model theoretic technology.

However, after initiating stability theory in the first place it seemed that the study of categorical structures had run its course (Baldwin-Lachlan theorem completely categorises theories which are categorical in powers). But recently Zilber realised that some of Shelah's abstract model theoretic technology regarding infinitary logics can be used to study concrete, well known, and very interesting mathematical structures.

For example, as John Goodrick mentions, if you try and axiomatize the interaction of the exponential function with the complex field i.e. you try and capture the theory of $\langle \mathbb{C},+ \cdot ,0,1, e^x \rangle$, and you want it to be categorical, then you need things like Schanuel's conjecture and the conjecture on the intersections of Tori (CIT) to hold. Along similar lines is the Zilber-Pink conjecture.

So model theory can give us some kind of justification as to why certain results should hold. For example, if you look the theory of the universal cover of a non CM elliptic curve over a number field, and you ask it to be $\aleph_1$-categorical, then it turns out that a famous theorem of Serre saying that the image of the Galois representation on the Tate module is open must be true.

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The connection between algebraic dynamics and the model theory of difference fields was first noticed by Chatzidakis and Hrushovski, who use it in a series of three papers entitled "Difference fields and descent in algebraic dynamics" to "prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line."

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Hi Alice. Nice to see you here. –  Andres Caicedo Jan 15 '11 at 4:41

Fesenko wrote here about "interactions of model theory, arithmetic and algebraic geometry and noncommutative geometry", but I don't remember if he mentiones Iwasawa theory.

A result coming from Y.I. Manin's idea to address the Mordell--Weil problem for cubic surfaces using model theorety are these reconstruction theorems.

BTW, as the André-Oort conjecture is an analogy to the Manin-Mumford conj. , and the later has been treated by model theoretic methods, applies model theory to the former (and common generalizations) too?

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Scanlon has done some work on this. The first link is to his paper, the second is to some slides that give an outline of the argument: math.berkeley.edu/~scanlon/papers/aoaug04.pdf math.berkeley.edu/~scanlon/papers/aocam2.pdf –  John Goodrick Dec 5 '09 at 16:27

There is the following nice result of Duesler and Knecht: "almost" every rationally connected geometrically integral variety over the maximal unramified extension of Q_p has a rational point. Here "almost" means that for a fixed Hilbert polynomial P(x), then for all sufficiently large primes p, every rationally connected variety over Q_p^{unr} admitting a projective embedding with Hilbert polynomial P has a Q_p^{unr}-rational point.

The proof takes a theorem of de Jong and Starr on rationally connected varieties over (F_p-bar)((t)) and transfers it via model-theoretic prestidigitation to Q_p^{unr} in much the same way that Ax-Kochen transfers the Greenberg-Lang result that F_p((t)) is C_2 to show that Q_p is "almost C_2". (In this case though whether there are any exceptions to the "almost" is still open, and apparently the experts -- still! -- believe that there should be none.)

See http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.2364v1.pdf

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If by "besides continutation of Hrushovski's work" you allow work of Hrushovski on other topics, you may be interested in the reading seminar which is documented by Terry Tao in his blog (first post here, there are 5 by now). It certainly falls under the heading "applied model theory".

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I don't know specifically about Iwasawa theory, but new applications of model theory to algebra and algebraic geometry were recently developed in a series of papers by Kazhdan and Hrushovki. For example, their paper Integration in valued fields is a top of an iceberg. Van den Dries has excellent notes on this paper on his website.

There is also a recent preprint by Moshe Kamenski Model theory and the Tannakian formalism on a different application of model theory.

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