Geometric stability theory is the principal part of the branch of model theory called geometric model theory?. It was introduced in works of Boris Zilber, Gregory Cherlin, Ehud Hrushovski, Anand Pillay, and others.
Geometric stability theory has largely to do with the model-theoretic classification of structures in terms of dimension-like quantities that can be axiomatized in terms of notions of combinatorial geometry such as matroid. Key guiding examples include vector spaces and algebraically closed fields, and many of the guiding concepts have an algebro-geometric flavor (e.g., Morley rank as a generalization of Krull dimension).
Such a structural geometric approach can make geometric stability theory an attractive key of entry into modern model theory for those mathematicians who are not already logicians.
Perhaps the key axiomatic notions of geometric stability theory (which at the outset don’t seem particularly wedded to logic) are those of pregeometry and geometry.
The monad is finitary, i.e., and , then there is a finite such that .
(Exchange condition) If , , and , then or . (Cf. matroid)
A geometry is a pregeometry such that and for all .
Let be a vector space, and let be the monad on whose algebras are vector subspaces of . Clearly is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus is a pregeometry.
Similarly, let be a projective space , and let be the monad on whose algebras are projective subspaces. Then is a geometry (the closure of a point is a point). Any pregeometry gives rise to a geometry in a similar way, in the sense that a pregeometry induces a geometry on the image of the function , , as explained in Remark 2.
Let be an algebraically closed field; let be the monad on whose algebras are algebraically closed subfields. Then is a pregeometry. That the exchange condition is satisfied is a classical result credited to Steinitz1.
Given a pregeometry , a subset is independent if for all , . An independent set said to be a basis for if . All bases of have the same cardinality, called the dimension of .
Given a pregeometry and a subset , there is a restriction pregeometry defined by the formula
It is immediate that is independent in iff it is independent in , and that it is a basis for in iff it is a basis for in . By this observation, to prove that any has a well-defined dimension, we may assume without loss of generality that . That a pregeometry has well-defined dimension was proven here.
There is a standard way of getting a geometry from a pregeometry . First, if , then replace by , equipped with the restriction pregeometry of the previous remark. Then define an equivalence relation on by if , and define a pregeometry on the quotient set by
where denotes the equivalence class of . This abstracts the process of taking a projectivization of a vector space.
For , an element is algebraic over if there is a formula and elements such that and
is finite. The algebraic closure of , denoted , is the set of elements of that are algebraic over .
The algebraic closure defines a finitary closure operator on .
That is monotone (preserves order) is obvious. Also the fact that is finitary is easy: given that satisfies for , then similarly for .
Taking to be the equality predicate , we see .
For idempotence of , suppose and where has exactly elements. Write down a formula that says has at most elements (by adding some extra inequalities, we can make this “exactly elements”):
Now for , let be a formula witnessing , i.e., where and has finitely many elements. Then
is a formula with parameters in that witnesses . This proves the idempotence of .
As before, this notion of algebraic closure can be restricted to a closure operator for a subset , via the definition for . One often writes just instead of , provided that is understood.
For an -structure , let be definable. is minimal if the only definable subsets of are finite or cofinite in . Slightly abusing language, if is a formula with parameters that defines , we also say is minimal. We say (or ) is strongly minimal if it is minimal in any elementary extension of .
A theory is strongly minimal if for any model of , the underlying set (definable by the formula ) is strongly minimal.
Let be an algebraically closed field. (Elimination of quantifiers, Chevalley’s theorem, etc.) Conclusion: ACF is a strongly minimal theory.
Divisible torsionfree abelian groups.
Non-example of dense linear orders.
(Baldwin, Lachlan) The algebraic closure operator on a minimal set is a pregeometry.
Let and suppose for ; we want to show . Thus, suppose is a formula with parameters from such that and , a finite number. As in the proof of Proposition 1, let be a formula that says . This is a formula with parameters from and is satisfied in .
If is finite, then since belongs to this set, would witness and we would be done. So suppose otherwise. Then this set is cofinite in , so that
for some finite , and we can again write down a formula with parameters from that says
If defines a finite subset of , then since is satisfied, we would reach the conclusion that , a contradiction. Hence defines a subset cofinite in . We may therefore choose elements such that is satisfied. By our supposition,
is cofinite for . Hence is inhabited, say by an element . We have at least elements such that is satisfied, namely . But now this contradicts the fact .
Anand Pillay, Geometric stability theory, Oxford Logic Guides 32
Misha Gavrilovich, Model theory of universal covering space of complex algebraic varieties, thesis, pdf
Although oddly enough, as explained at the MacTutor biography page, what is called the Steinitz exchange condition was set out by Ernst Steinitz in a 1913 publication on convergent series and apparently not (?), as might be supposed, in his Algebraische Theorie der Körper (Crelle’s Journal, 1910). His 1913 lemma was for vector spaces. ↩