Is there any intersection between the ideas of Algebraic Topology/Geometry (I know that there is most certainly a non-trivial intersection between Algebraic Geometry, Algebraic Topology, Arithmetic Geometry, etc.) and the ideas of Model Theory, Set Theory and more fundamental topics? I have heard that there are some powerful tools from commutative geometry being applied to topology (correct me if I'm wrong of course) like Andre-Quillen Cohomology, and I remember once seeing a talk about applying some ideas from model theory to commutative algebra (something like applying model theory to put upper bounds on Betti numbers of Cohen-Macaulay rings?). So I'm just wondering if any of these foundational ideas are ever relevant (in the non-obvious ways, i.e. we can't do math without foundations) to these other subjects.
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3$\begingroup$ Topos theory might be something you would be interested in (See Moerdijk and Mac Lane: Sheaves in Geometry and logic). $\endgroup$– Thomas RotCommented Apr 14, 2011 at 16:10
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$\begingroup$ o-minimal structures $\endgroup$– yoyoCommented Apr 14, 2011 at 20:32
2 Answers
Yes. The area of model theory you are asking about is known as "Geometric stability theory". This is a very active area of research. The standard reference is the book by Pillay with the same name.
These connections have actually gathered a fair amount of recent attention (a google search will provide you with links to conference webpages and some articles), one of its highlights being Hrushovski's "The Mordell-Lang conjecture for function fields", J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. But, really, most recent work in model theory is in this area. For example, take a look at the list of publications by Scanlon, for many interesting examples of model theoretic ideas leading to number theoretic results.
Actually, connections between model theoretic ideas and algebraic geometry have been around for a while, starting with work of Abraham Robinson, though it is fair to say that their recent sophistication is due to the deep insights of Hrushovski, Pillay, Zilber, and their students and collaborators. (I fear any list of names I give is bound to be embarrassingly incomplete.)
Similarly, model theoretic work on "$o$-minimality" is connected in serious ways with real-algebraic geometry. There have also been some interesting connections between this area and set theory, mostly due to the fact that real-algebraic geometry gives us some insight on the study of rings of continuous functions and their quotients. You may want to look at "Super-real fields. Totally ordered fields with additional structure", London Mathematical Society Monographs. New Series, 14. Oxford Science Publications; The Clarendon Press, Oxford University Press, New York, 1996, by Woodin and Dales.
In a different direction, Talayco, a student of Andreas Blass, studied connections between cohomology and set theory.
For a completely different approach, though perhaps not exactly in the direction you intend, locales and, in general, topos theory, allow a foundational presentation that people feeling more comfortable with category-theoretic ideas may prefer to the set theoretic approach. The usefulness of the framework lies in part in that it gives us a way to study duality theorems (such as the one relating Stone spaces with Boolean algebras) in a unified fashion. See for example, "Stone spaces" by Johnstone or "Natural dualities for the working algebraist" by Clark and Davey.
There is a lot available, from the pioneering work of Ax-Kochen to the more recent spectacular work of Ehud Hrushovski. I would not agree with the implication in the post that Model Theory, Set Theory are more fundamental than, say, Algebraic Geometry. Equally fundamental?
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$\begingroup$ Perhaps. I always sort of thought that set theory, model theory, logic and proof theory were supposed to be the so-called foundations of math. $\endgroup$ Commented Apr 14, 2011 at 16:15
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$\begingroup$ @JBeardz: For contingent historical reasons, much of pure mathematics is now, at least in principle, expressed in set-theoretic language. It very well could have been category-theoretic language. And one could even argue, whimsically, that Model Theory is a branch of Algebraic Geometry. $\endgroup$ Commented Apr 14, 2011 at 16:38
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$\begingroup$ @AndréNicolas, As if the model theory is purer. In model theory, we have logical formulas for definable sets, and analogously, in algebraic geometry, we have equations for algebraic varieties over a given field. $\endgroup$ Commented Apr 30 at 13:27