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I know very basic model theory (compactness, Lowenheim-Skolem, EF games, at that level), and I'd like to pick up more, mostly out of intrinsic interest and partially because I think it would give me an interesting perspective as I dive into my main subject, which is algebraic geometry.

I've been told that the best source for this sort of thing is Marker. After flipping through it I'm kind of confused about how connections between the subjects arise. Are the applications of model theory to algebraic geometry strictly classical? Are nonclassical applications too advanced to show up in something like Marker?

I ask because there's no mention at all of sheaves in Marker and nothing really about categories, and from a little googling this doesn't seem to be unusual. I'm not one to demand categorification for no reason but these are how the basic objects of modern a.g. are defined so I would expect them to show up in applications from a closely linked subject.

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  • $\begingroup$ Maybe doesn't precisely fit mainstream "model theory" - but often, a topos (for example the category of sheaves of sets on a topological space, or more generally on a Grothendieck site) is viewed as a model of an "intuitionistic type theory". $\endgroup$ – Daniel Schepler Aug 1 '18 at 21:43
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    $\begingroup$ One addendum to the fine answer below: o-minimal geometry has strong global finiteness properties which greatly restricts the topologies that you can put on spaces to have something like a definable sheaf of modules. For example, if you want to consider the sheaf of definable functions on some space, your space cannot have infinitely many disjoint open subsets (else you may put a bump function on each open and get a definable set with infinitely many connected components). So already you're looking at something which is much closer to the Zariski topology on a variety. $\endgroup$ – KReiser Aug 2 '18 at 5:36
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    $\begingroup$ For some recent nice work in algebraic geometry which uses model theory, you may wish to look at Pila's results on the Andre-Oort conjecture (people.maths.ox.ac.uk/pila/OminimalAO.pdf), or Aizenbud-Avni on representation growth (arxiv.org/pdf/1307.0371v2.pdf). $\endgroup$ – KReiser Aug 2 '18 at 5:41
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You have the right impression: most applications of model theory to algebraic geometry are essentially "classical", in the sense that they are about varieties as definable sets over algebraically closed fields, not (explicitly) about sheaves and schemes. The model theoretic approach is actually very similar to the view of the foundations of algebraic geometry promoted by André Weil in the 40s (the monster model is essentially Weil's universal domain) which was largely superseded by Grothendieck's approach not long after.

One obstruction to the use of sheaves in model-theoretic algebraic geometry is that it is the constructible topology, not the Zariski topology, which is most natural in model theory. Here algebraic sets are clopen, not just closed - this corresponds to the fact that first-order languages are closed under complement (negation) and projection (quantification). Of course, the constructible topology is totally disconnected, which removes much of the geometric content captured by sheaves.

On the other hand, model theory has proven to be very useful in fields adjacent to algebraic geometry, where a good theory of schemes is not available or is much more complicated than in the classical case. I'm thinking of semialgebraic geometry (and o-minimal generalizations), differential algebra, difference algebra, Berkovich spaces, etc.

That's not to say that schemes and sheaves are nowhere to be found in model theory - I'll leave it to someone else to give some references to some places where they appear. For my part, I'll link you to Angus Macintyre's paper Model Theory: Geometrical and Set-Theoretic Aspects and Prospects from 2003. Here Macintyre surveys the history of model theory and suggests (in a somewhat vague way) a future in which model theory has more in common with Grothendieck-style algebraic geometry. I think this paper has been fairly influential, but the revolution hasn't arrived yet.

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