fea-pillay.qxp 10/16/00 9:03 AM Page 1373

Model Theory Anand Pillay

odel theory is a branch of mathemat- pathology. This is of course true in many ways. In ical logic. It has been considered as a fact, possibly the deepest results of twentieth- subject in its own right since the 1950s. century logic, such as the Gödel incompleteness I will try to convey something of the theorems, the independence of the continuum model-theoretic view of the world of hypothesis from the standard axioms of set theory, mathematicalM objects. The use of the word “model” and the negative solution to Hilbert’s Tenth in model theory is somewhat different from (and Problem, have this character. However, from al- even opposed to) usage in ordinary language. In most the beginning of modern logic there have the latter, a model is a copy or ideal representation been related, but in a sense opposite, trends, often of some real object or phenomena. A model in the within the developing area of model theory. In the sense of model theory, on the other hand, is 1920s Tarski gave his decision procedure for supposed to be the real thing. It is a mathematical elementary Euclidean geometry. A fundamental structure, and that which it is a model of (a set of result, the “compactness theorem” for first-order axioms, say) plays the role of the idealization. logic, was noticed by Gödel, Mal’cev, and Tarski in Abraham Robinson was fond of the expression the 1930s and was, for example, used by Mal’cev “metamathematics” as a description of mathemat- to obtain local theorems in group theory. Abraham ical logic. His dream, which he realized in many ways, Robinson in the 1950s introduced differentially was to show how metamathematical arguments and closed fields and discovered nonstandard analysis. considerations can produce new mathematical The eve of the modern era saw James Ax’s decid- results. Logic began as the delineation and classifi- ability for the theory of finite fields and Ax-Kochen’s cation of (logically) valid forms of reasoning. One asymptotic solution to a problem of E. Artin. The might not expect this science of tautology to have modern era has seen, often for unexpected reasons much to say about the world. However, for reasons and sometimes because of the internal development people do not fully understand, it does have some- of the subject, that model theory is in a position thing to say about the mathematical world. to discern amazing patterns and analogies in tame Logic is, in the popular imagination, often mathematics (those areas of mathematics not sub- associated with undecidability, paradoxes, and ject to Gödel undecidability and incompleteness) and in the process to obtain new results. This is Anand Pillay is Swanlund Professor of Mathematics at the what I want to discuss. University of Illinois, Urbana-Champaign. His e-mail I will have to introduce some technical mater- address is [email protected]. He was partially ial, outlining the basic notions and objects of model supported by NSF Grant DMS 00-70179. theory as we see them now. In the process I will This article was influenced by a series of talks given refer back to and explain something of the older by the author at the Southwest Center for Arithmetic results mentioned above, but my aim is rather to Algebraic Geometry in May 2000, and he thanks the number theorists at the University of Arizona, Tucson, for describe current work and prospects for the future. their hospitality. He thanks also Enrique Casanovas and I will be limiting my description and analysis to a Ward Henson for their helpful comments on an earlier few “main trends” in model theory. As such I will draft of this article, as well as the editors for their either not touch on or give scant attention to many encouragement and hard work in bringing it to fruition. interesting fields of research, such as finite model

DECEMBER 2000 NOTICES OF THE AMS 1373 fea-pillay.qxp 10/16/00 9:03 AM Page 1374

theory (definability in finite structures, which is re- We want to be able to distinguish clearly between lated to both database theory and computational labels and the objects to which they refer. complexity), universal algebra, nonstandard analy- Given a structure M, say, let us introduce labels sis and the model theory of infinite-dimensional or names for the distinguished functions and sets. spaces, infinitary logic, generalized quantifiers, This choice of labels gives rise to a vocabulary or “abstract” model theory, and models of arithmetic language for the structure M. In fact, the usual way and set theory. of introducing structures is to start with a vocabu- For further reading on model theory in the spirit lary L = (fi,Rj )i,j consisting of function labels and of this article, I recommend starting with the set labels of specified arities (i.e., the integers n volume [4]. A classic text on model theory is [2]. An mentioned above) and then to define an L-structure elegant and very accessible text on o-minimality is M to be a set X equipped with distinguished M M [3]. For more on the connections with diophan- functions fi and sets Rj corresponding to these tine-geometric issues, I recommend [1] and [5]. labels. One often says, for example, that f M is the interpretation of the label f in the structure M. Structures and Definability Different structures can have the same vocabulary. The key notion of model theory, which I think For example, suppose our language L consists cannot be avoided even though there are many of a single binary function label f. We can form an attempts in popular expositions to get around it, L-structure M whose underlying set is the integers Z M is the notion of “truth in a structure”. A structure such that the interpretation f of f is M here is simply a set X, say, equipped with a addition of integers. Another L-structure N is Z distinguished family of functions from Xn to X obtained by again taking as the underlying set but defining f M to be multiplication of integers. (various n) and a distinguished family of subsets 0 of Xn (various n). Here Xn is the Cartesian For another example, suppose the language L consists of a single set label R of arity 2. We can product X X, n times. We shall assume 0 that the diagonal {(x, x):x ∈ X} is among the form an L -structure P whose underlying structure Z P distinguished sets even if it is not explicitly men- is the integers such that the interpretation R of R is the relation set for <, i.e., the subset of tioned. There is not much to say here except that (x, y) ∈ Z2 with xfield, the algebraic geometer will be interested in ∧ symbols for the usual logical operations “and” ( ), the algebraic subvarieties of X, X X, etc., among “or” (∨), “implies” (→), “not” (¬), “there exists” (∃), which of course is the diagonal. So the important “for all” (∀), as well as an infinite supply of thing in looking at a structure is not only the “variables” (x,y,...), we can build up first-order underlying set X but also the family of functions expressions that can be interpreted as making and sets with which it is equipped. The integers statements about the structure M and/or about considered as an additive group (i.e., with addition elements of that structure. The first-order aspect the only distinguished function) is a very different is that the variables range over elements of the structure from the integers considered as a ring. underlying set X (rather than subsets). It should A slightly more general and mathematically quite also be emphasized that a first-order formula is a natural notion is that of a many-sorted structure. In finite string of symbols. the place of a single underlying set or universe X, we What about truth? Suppose f is a label for a unary have an indexed family (Xi)i of universes. The function. Consider the first-order expression : distinguished relations and functions live on or go ∀x∀y(f (x)=f (y) → x = y). between various Cartesian products of the Xi. Every- thing we say below, including the notion of truth In words this expresses that whenever f (x)=f (y), of first-order formulas in a structure and the then x = y. The expression is true or false in M compactness theorem, adapts unproblematically to according to whether or not the function f M from the many-sorted setting. X to X is one-one. This is an example of a first- Nevertheless, let us work with a single underlying order sentence. On the other hand, the expression set X. If fi and Aj are the distinguished functions (x): and sets for a structure M , we write M as ∃y(f (y)=x) (X,fi,Ai)i,j. The additive group of reals becomes (R, +, 0) in this notation. Here + is the addition has a “free” or “unquantified” variable x. Whether function from R2 to R, and 0 refers either to the it is true or false in M depends on what x is. The constant function from R0 to R or to the singleton expression (x) will be true of an element a ∈ X set {0}. But we are getting ahead of ourselves. if and only if a is in the range of f M. We say that

1374 NOTICES OF THE AMS VOLUME 47, NUMBER 11 fea-pillay.qxp 10/16/00 9:03 AM Page 1375

(x) is an example of a first-order formula with free sentences true in all members of a certain class of variable x. Traditional notation is: structures. In the case of the class of all finite fields, decidability was proved by James Ax (1968). M |= for “ is true in M”. In ordinary So we have decidability results for fields sur- language “M is a model of ”. If is a rounding number theory. Number theory itself possibly infinite set of sentences, we say lives directly in the structure (Z, +, ), and Gödel’s M |= , M is a model of , if it is a work yielded undecidability of the theory of this model of each ∈ . structure. In fact, Matijasevich’s negative solution Regarding the formula (x) above, we write of Hilbert’s Tenth Problem says that there is a M |= (a) for “(x) is true of a in M”. fixed polynomial P(X1,...,Xn,Y1,...,Ym) over Z It is important to understand not only the power such that there is no effective procedure for de- of first-order expressibility but also its limitations. termining, for any given a1,...,am ∈ Z, whether Consider a structure M of the form (R,f,<), where or not the equation P(X1,...,Xn,a1,...,am)=0 f is a function from R to itself and < refers to the has a solution in Z. Gödel’s coding methods show relation subset for “less than” in R2 . Let a ∈ R. that all “effective mathematics” is coded in or Then there is a first-order formula (x) such that contained in the structure (Z, +, ). for any a ∈ R, M |= (a) if and only if f is contin- Related to decidability issues is the question of uous at a. Specifically (x) informally is: identifying axioms for theories of specific struc- tures. For example, it was proved by Tarski (1949) For all y and y such that 1 2 that the theory of the field C is axiomatized by the y numbers is One wants to express that for every x, either x =0 elementarily equivalent to (C, +, ), as the latter or x + x =0 or x + x + x =0 or …. An infinite dis- contains transcendental elements. As in the issue junction appears inside the universal quantifier, so of torsion abelian groups discussed above, the this is not a first-order sentence. A rigorous proof limitations of first-order expressibility are of nonexpressibility could be based on the com- responsible. If one tries to write down a sentence pactness theorem discussed in the next section. expressing the existence of a transcendental A key feature of introducing labels is that we can element, one inevitably finds oneself writing ∃x compare in various ways different structures that followed by an infinite conjunction. have the same labels (vocabulary). The notion of Deeper results along these lines, but for isomorphism is clear, but we also have the “Henselian valued fields”, were obtained by James a priori weaker notion, elementary equivalence. Ax and Simon Kochen (1965, 1966) and Yuri L. Two structures M and N with the same vocabulary Ershov (1965). A valued field K is a field equipped are said to be elementarily equivalent if the first- with a map v from K into ∪ {∞}, where is an order sentences true in M are precisely the ones true ordered group, the restriction of v to the multi- in N. The elementary or first-order theory of a plicative group K is a homomorphism, v(0) = ∞, structure M is by definition the set of first-order and v(x + y) min{v(x),v(y)} . A valued field is sentences true in M. This is exactly what we call a Henselian if it satisfies “Hensel’s Lemma”, whose complete theory. statement we do not need and consequently omit. Classical decidability results can now be stated It is convenient to think of a valued field as a and made sense of. Tarski (1951) proved that the “composite” of three structures—the underlying elementary theory of the real field (R, +, , 0, 1) is field (K,+, ), the value group ( , +,<), and the decidable. That is, there is an effective procedure “residue field” (k, +, )—together with the valua- for deciding whether or not a given first-order tion v : F → ∪ {∞} and the residue map from sentence in the appropriate vocabulary is true the valuation ring of K to k. The Ax-Kochen-Ershov in the structure. For example, may express that result says that if we restrict ourselves to Henselian a given finite set of polynomial equations in valued fields of characteristic 0 whose residue indeterminates X1,...,Xn over Z has a solution in fields are of characteristic 0, the theory of this Rn. But there are more complicated examples: for composite structure is determined by the theory example, whether the set of solutions in Rn of of the value group and the theory of the residue such a system has N connected components can field. also be decided, for nontrivial reasons. Often the proofs of decidability and axiomati- Decidability is known also for the theory of the zability results yield information on the “definable fields C of complex numbers and Qp of p-adic sets” in a structure. In fact, identification of the numbers. One can consider also the set of definable sets in a structure has become a central

DECEMBER 2000 NOTICES OF THE AMS 1375 fea-pillay.qxp 10/16/00 9:03 AM Page 1376

focus. So we should explain what these definable or phases. The first is to understand the category sets are. D(M) and takes the form of identifying a man- If a structure M has underlying set X and vo- ageable class of first-order formulas such that cabulary L, and if an L-formula (x1,...,xn) is every definable set is defined by one of these for- given, the set M defined by in M is by defini- mulas, as in the examples of the real, complex, and n tion the set of (a1,...,an) ∈ X such that is true p-adic fields above. Such a procedure is called of (a1,...,an) in M. The definable sets in M are pre- quantifier elimination. The second aspect, under- cisely the sets M. A function f from the definable standing quotients, is rather more complicated. In set Y to the definable set Z is said to be definable the best of all possible worlds, for any Y/E as if its graph is. Each member of the original family above, there exist a set Z in D(M) and a definable of distinguished sets and functions is definable. surjection f from Y to Z such that f (x)=f (y) if and The reader can verify that the logical connectives only if E(x, y). If this happens (as it does for the translate into the closure of the class of definable structures (C, +, ) and (R, +, )), we say that M has sets under various set-theoretic operations: for elimination of imaginaries. example, complementation, finite unions and intersections (in a given ambient space Xn ), and Compactness finite Cartesian products. Also, the image of a de- The compactness theorem says that if every finite finable set under a definable function is definable subset of a set of sentences has a model, then (using the existential quantifier). We obtain from has a model. Equivalently, if the sentence is a the structure M a category D(M), the category of logical consequence of the set of sentences, then definable sets and functions, which can also be de- is already a logical consequence of a finite fined syntactically from (and thus only depends on) subset of . What was at the time considered a the theory of M. There is a somewhat richer cate- rather surprising implication of this was Ax’s 0 gory D (M) containing, in addition to the definable injectivity-implies-surjectivity theorem: any sets, the fibers of definable functions. This is the injective morphism from a complex algebraic category of sets definable in M with parameters. variety V to itself is surjective. The proof goes as Let us consider the case of the structure follows. Suppose for a contradiction that f : V → V R ( , +, , 0, 1). The natural ordering xtuple (X1,...,Xn) of indeterminates. in which we exhibit the coefficients a¯. Similarly f Tarski’s quantifier elimination theorem (1951) is defined by a system F(¯x, a¯) of polyomials. We can says that the definable sets are precisely finite now find a first-order sentence saying: there is unions of sets defined by y¯ such that F(, y¯) defines an injective but non- P(X) > 0 ∧ Q1(X)=0∧∧Qm(X)=0, surjective map from the set defined by P(¯x, y¯) to itself. The sentence is true in (C, +, ), hence is where P and the Qj are polynomials over the ra- a consequence of the axioms for algebraically Q tionals . The sets definable with parameters are closed fields of characteristic 0. By compactness, just the same thing except that the coefficients in is true in (Fp, +, ) for some (in fact almost all) P and the Q can be arbitrary in R. These are pre- j primes p, where Fp is the algebraic closure of the cisely the “semialgebraic sets” in the subject of al- field with p elements. We get a contradiction by gebraic geometry over R. The content of the the- counting. In fact, the algebraic closure K of Fp is orem is that the projection of a semialgebraic set locally finite. So if V is a variety over K and f is an Rn+1 Rn from to is also semialgebraic. In any case, injective morphism over K from V to V, then for what we obtain are precisely the objects of “semi- each finite subfield F of K over which V and f are algebraic geometry”. Similarly the sets definable in defined, the restriction of f to V(F) is onto. As V C Q ( , +, , 0, 1) are precisely the -constructible sets is the union of all such V(F), f : V(K) → V(K) is (Chevalley). The definable sets in the p-adic field onto. Q ( p, +,.,0, 1) are the sets quantifier-free definable A similar proof yields Lazard’s theorem that an after adding relations for the nth powers, as proved algebraic group with underlying variety affine by Macintyre (1976). n-space is unipotent. This general method gives, In the general case it is natural to “complete” among other things, a rigorous formulation of D(M) by adjoining quotient sets of the form Y/E, “reduction mod p” technology. where Y is definable and E is a definable Another implication with a similar flavor, equivalence relation on Y. We call this completion eq eq although now using the deep Ax-Kochen-Ershov D (M) or D(M ). As the latter notation suggests, analysis of the elementary theories of Henselian this is the category of definable sets in a certain valued fields mentioned above, is the asymptotic many-sorted structure Meq attached to M. It is solution to a conjecture of E. Artin, as follows: now recognized for model-theorists that to un- derstand a given structure M is to understand the Theorem 1. For any given d, for all sufficiently category Deq(M). In practice this has two aspects large primes p, for all n>d2, every homogeneous

1376 NOTICES OF THE AMS VOLUME 47, NUMBER 11 fea-pillay.qxp 10/16/00 9:03 AM Page 1377

polynomial over Qp in n variables of degree d has nonstandard models are the normal place to study a nontrivial zero in Qp. definability. These are “universal domains” for a theory. Such an object depends on the choice of a The actual conjecture was the statement above suitable (usually uncountable) cardinal . We but with “for all primes” replacing “for all suffi- will restrict our attention to the case where is ciently large primes”. chosen to be the first uncountable cardinal. So Here is a sketch of the proof of Theorem 1. It given, say, a complete, countable theory T, by a is enough to prove the theorem for the case universal domain for T we mean a model M of T n = d2 +1. Let be a first-order sentence in the such that every countable model of T elementar- language of rings expressing that every homoge- ily embeds in M, uniquely up to an automorphism neous polynomial of degree d in d2 +1 variables of M. (These are the “homogeneous-universal has a nontrivial zero. By a theorem of Lang, models” of Morley-Vaught.) In fact, the field of is true in the field Fp((t)) of quotients of complex numbers (in no way a nonstandard object) (*) the ring of formal power series over the is a universal domain in this sense for the theory of algebraically closed fields of characteristic 0. finite field Fp, for all primes p. Sometimes by abuse of language we will say “sat- Let be the set of sentences in the language for urated model” in place of universal domain. valued fields consisting of Saturation amounts to saying that any countable 1. the axioms for Henselian valued fields of char- family of definable subsets of Mn, every finite acteristic zero whose residue fields have char- subset of which has nonempty intersection, has acteristic zero; nonempty intersection. In the context of alge- 2. for the value group, the sentences true in braically closed fields, this provides the existence (Z, +, 0,<); of generic points for varieties. In fact, our notion 3. for the residue field, the condition “charac- of universal domain is exactly that used by Weil in teristic zero” (the infinite set of sentences ex- algebraic geometry (and later by Kolchin in dif- pressing that the characteristic is different ferential algebraic geometry), although it has from p for each prime p), together with the somewhat gone out of use, replaced by the scheme- theory of finite fields. theoretic point of view. From ( ) together with the Ax-Kochen-Ershov o-Minimality and Real Geometry result, one deduces with not much difficulty that As noted above, the sets definable with parame- is a logical consequence of . By compactness, 0 ters in the structure (R, +, , 0, 1,<) are precisely is a logical consequence of a finite subset of the semialgebraic sets. Good properties of (the . For all sufficiently large p, the p-adic valued 0 class of) semialgebraic sets include these: field Qp is a model of , thus of , and the proof is complete. a. any semialgebraic set has finitely many con- The compactness theorem is fundamental in first- nected components, order model theory. To start with, it gives rise to b. a compact semialgebraic set admits a semial- “nonstandard models”: any infinite structure has a gebraic triangulation, proper “elementary extension”; in fact, it has ele- c. any semialgebraic family of semialgebraic sets mentary extensions of arbitrarily large cardinality. contains only finitely many sets up to semial- An extension N of a structure M is a structure in the gebraic homeomorphism. same vocabulary such that for each n-ary relation One can ask whether similar properties hold for the symbol R, RM is precisely RN ∩ Mn, etc. It is fur- sets definable in the structure (R, +, , 0, 1,<,exp) thermore an elementary extension if every sentence = Rexp, where exp is the real exponential func- with parameters from M that is true in M is also true tion. Tarski was the first person to ask about the in N. Nonstandard analysis, invented by Abraham decidability of the theory of Rexp. Hovanskii showed Robinson, takes M to be a suitably rich structure for that the zero set of an “exponential” polynomial has doing analysis; constructs a nonstandard model M; finitely many connected components. Subsequently proves theorems in M using, for example, the ex- Alex Wilkie (1996) proved that any subset of Rn de- istence of “infinitesimals”; and then transfers the finable in Rexp is the projection of an exponential result back to the standard structure M. A related variety in some Rn+m, namely, is subexponential. area, called “models of arithmetic”, uses nonstandard The content of this is a “theorem of the comple- models of first-order Peano arithmetic, or of weaker ment”: the complement of a subexponential set is axiom systems, to study the strength of such axiom also subexponential. In any case, a consequence is systems. Among the successes of this kind of work that (a) holds for definable sets in Rexp. was a new “natural” Ramsey-style combinatorial fact Already in the early 1980s it was realized by that is true but not provable in Peano arithmetic van den Dries (1984) and by Steinhorn and me (Paris-Harrington). (1984) that there is a general model-theoretic con- From a general model-theoretic point of view text here. What we called an o-minimal structure (and in many concrete examples), certain is a structure M =(X, <, Ai)i, where < is a dense

DECEMBER 2000 NOTICES OF THE AMS 1377 fea-pillay.qxp 10/16/00 9:03 AM Page 1378

linear ordering on X and the Ai are arbitrary rationality taken with respect to the Grothendieck subsets of various Xn satisfying: every subset of ring of algebraic varieties over C. X definable with parameters in M is a finite union of points and intervals (a, b) with a ∈ X ∪ {∞} Interpretability and Invariants and b ∈ X ∪{+∞}. The ordering gives a topology In the article so far I have mostly described appli- on each of X,X X,.... One may want to restrict cations involving usually a detailed model-theoretic oneself to the case that X is the real line with the analysis of specific structures (axiomatizing the usual ordering. In any case, from the assumption theory, describing definable sets), together with a of o-minimality, a condition on parametrically sometimes ingenious use of elementary model- definable sets in 1-space, was deduced a host of theoretic techniques (like compactness). The work properties of arbitrary definable sets. For example: in the previous section is a bit different in that it any definable subset of X X has finitely is informed by the model-theoretic techniques many definably connected components, definable from the general theory of o-minimality. In any functions are piecewise continuous, and much case, one might get the impression that model more. The notion and theory of o-minimality can theory is simply a collection of techniques and with hindsight be considered to be a response to concepts that come to life only in clever applica- Grothendieck’s challenge to investigate classes of tions. In fact, there exists a subject “model theory sets with the tame topological properties of the for its own sake” that is quite developed but in semialgebraic sets. some sense is still in its infancy. Moreover, there What Wilkie actually proved in 1996 was the is a growing collection of applications in which the o-minimality of Rexp. From the general theory of more sophisticated aspects of this theory play an o-minimality one concludes that all the topologi- increasingly important role. This section is cal-geometric properties of semialgebraic sets devoted to exploring what is or could be “model (including (a), (b), and (c) above) pass over to theory for its own sake”. definable sets in Rexp. The main theme of current In areas of mathematics such as geometry and work is to include richer and richer classes of topology (and many others), a given class of functions and sets in o-minimal structures on R. objects, such as manifolds, algebraic varieties,…is It was noticed fairly early that the class of “finitely identified, together with an appropriate notion of subanalytic sets” (subanalytic when viewed in isomorphism (“being the same”), and tautologi- projective space) is subsumed by o-minimality. cally a central aim of the area becomes the classi- Wilkie (2000) has more recently shown that the fication of these objects up to isomorphism. In class of “Pfaffian functions” belongs to an o- most cases this classification will never be fully minimal structure. It is hoped that the second part realized, but the attempt to do a classification, of Hilbert’s Sixteenth Problem (uniform bounds the techniques developed, and more importantly on the number of limit cycles of planar polynomial the connections discovered between the various vector fields) can be approached by finding a suit- classification problems in different areas are part ably rich o-minimal structure on R. A problem of of the process of uncovering and learning more Hardy on asymptotics was solved by studying a about the mathematical world. Some of these certain nonstandard model of Rexp, the field of endeavors can be seen exactly as classifying the LE-series (van den Dries, Macintyre, and Marker, definable sets in a given structure. For example, the 1994). Issues raised in this paragraph can be seen classification of complex algebraic varieties up to as part of the special theory of o-minimality. There birational isomorphism is more or less the same is also a general theory of o-minimality, concerned thing as the classification of sets definable with with arbitrary o-minimal structures. A deep result parameters in the single structure (C, +, ) up to by Peterzil and Starchenko (1998) essentially definable bijection, although on the face of it this reduces this to the study of o-minimal expansions translation is not particularly helpful. of real closed fields. In any case, central objects of (first-order) model Analogies, model-theoretic or otherwise, be- theory are structures and their first-order theo- tween the real and p-adic fields have always been ries, as well as possibly incomplete theories. What productive. There is as yet no general theory that is the notion of sameness for structures? As these bears the same relation to the p-adic fields that structures (and theories) may be in different o-minimality bears to the real field. Nevertheless, vocabularies, isomorphism does not make sense. the understanding of definable sets in the p-adic The right notion of sameness is “bi-interpretabil- fields, as well as expansions by analytic functions, ity”. Let us first consider structures. Suppose M and has led to spectacular results. I mention Jan Denef’s N are structures in possibly different vocabularies. proof (1984) of the rationality of certain Poincaré An interpretation of M in N is a bijection f of the series, where the coefficients are on the face of it underlying set of M with a definable set in Neq (so definable with an existential quantifier. This has possibly a quotient object), such that for every been generalized by Denef and Loeser (1999) with definable set Y in M, f (Y ) is definable in N. M and Qp replaced by the valued field C((t)) and with N are bi-interpretable if there are interpretations

1378 NOTICES OF THE AMS VOLUME 47, NUMBER 11 fea-pillay.qxp 10/16/00 9:03 AM Page 1379

f of M in N and g of N in M such that the map f g called geometric stability theory by posing the is definable in N and g f is definable in M. One problem of classifying uncountably categorical can similarly define interpretations and bi-inter- theories up to bi-interpretability. Zilber pointed out pretability with parameters. As an example for any the role and importance of the nature of definable field F, the structures (F,+, ) and the affine plane groups. This observation has led to the study of over F considered as an incidence structure (points, simple uncountably categorical groups, motivated lines, and incidence) are bi-interpretable (with by the conjecture (Cherlin-Zilber) that they are parameters). Bi-interpretability of structures M precisely the simple algebraic groups over alge- and N translates into a syntactic (at the level of first- braically closed fields. order formulas) bi-interpretability of the theory The issue of classification up to bi-inter- of M with the theory of N (a notion that the reader pretability has led also to important interactions is free to formulate). So, tautologically, the classi- with permutation group theory. Let us restrict our fication of first-order structures and/or theories up attention to countable structures M with the to bi-interpretability is a central issue. This links feature that the automorphism group Aut(M) of up with our earlier insistence on the category of M has finitely many orbits on n-tuples for all n. definable sets D(T ) of a theory T: the bi-inter- Such a structure M will be the unique countable pretability of complete theories T1 and T2 roughly model of its theory. Aut(M) becomes a topologi- translates into an equivalence of categories cal group when we stipulate that the stabilizers of eq eq between D (T1) and D (T2). finite tuples are basic open subsets. Then the All of this is a triviality, but once expressed it bi-interpretability type of M is determined by explains and unifies an enormous amount of work Aut(M) as a topological group, and one issue is on the pure side of model theory. In particular the when and whether this can be recovered from the issue of finding interesting and useful invariants abstract group Aut(M). for the bi-interpretability type of structures/ theories is immediately put on the table. Stability and Diophantine Geometry Historically the development of “model theory A structure M is said to be unstable if it interprets for its own sake” focused on a certain invariant IT . a bipartite graph (P,Q,R) with the feature that for This IT is the function that associates to a cardi- each n there are ai ∈ P and bi ∈ Q for i =1,...,n nal the number of models of T of cardinality , such that R(ai,bj ) if and only if iSaharon Shelah and solved in the early 1980s of classifying the possible functions IT , it was by him. The solution appears in the second edition natural to focus on stable theories. Stability is a of his book Classification Theory and the Number very strong property. There are few natural ex- of Non-isomorphic Models (1990). He considered amples of stable structures: abelian groups (G, +), only countable complete theories T and the re- algebraically closed and separably closed fields striction of IT to uncountable cardinals. Early work (K,+, ), differentially closed fields (K,+, ,D). was done by Morley, who proved that if IT ()=1 More recently it was realized that compact com- for some uncountable , then it equals one for all plex manifolds are also stable; the structure on uncountable . These theories T of Morley’s, as the compact complex manifold X consists of the well as their models, are called uncountably cate- analytic subvarieties of X, X X,…. On the other gorical. Shelah produced an amazing body of work, hand, typically the structures considered in earlier in which a number of other invariants, such as sections, such as the real field and p-adic field, are stability, superstability,…, were introduced, as well unstable. as a theory of how complicated structures are built Stability can be recognized also by looking at the up from simple structures (geometries) by what Boolean algebra of definable subsets of a structure. geometers will recognize as fibrations. Henceforth by “definable” we will mean definable The archetypal example of an uncountably cat- possibly with parameters. Given M and an egorical theory is the theory of algebraically closed L-formula (x, y¯), we obtain the Boolean algebra fields of some fixed characteristic: the uncountable of subsets of M defined by finite Boolean combi- models are classified simply by their cardinality, nations of formulas (x, a¯), a¯ a tuple from M. This which equals the transcendence degree over the Boolean algebra has an associated Stone space, prime field. Boris Zilber invented what is now namely the space of ultrafilters. T is stable if and

DECEMBER 2000 NOTICES OF THE AMS 1379 fea-pillay.qxp 10/16/00 9:03 AM Page 1380

only if for any countable model M of T and X is linear or there is a definable minimal field L-formula (x, y¯), the associated Stone space is nonorthogonal to X. This is false in general, but its countable. The issue of the complexity of Stone truth in a certain axiomatic context (Zariski struc- spaces of Boolean algebras of definable sets is tures), as proved by Hrushovski and Zilber (1996), pervasive in stability theory and is one route to is of great interest for a number of examples. I defining various “dimensions” of definable sets. For will not give a definition of Zariski structure other example, given a saturated structure M and a de- than to say it is a minimal set in which certain finable set X in M, we will call X minimal (intuitively definable sets are identified as being “closed” and 1-dimensional and irreducible) if the Stone space some intersection-theoretic assumptions are made. of the Boolean algebra of all definable subsets of In any case, Zilber’s conjecture is true for minimal X has a unique nonisolated point or, equivalently, definable sets in separably closed fields, differen- if X is infinite and any definable subset of X is fi- tially closed fields, and compact complex manifolds. nite or cofinite. Among the general theorems about modular In the structure (C, +, ) the minimal sets are the definable sets (Hrushovski-Pillay 1987) is: algebraic curves (up to finitely many points). In the Theorem 2. If X is a modular definable group, more general structure of compact complex man- then every definable subset of Xn is, up to finite ifolds the minimal sets are essentially the simple Boolean combination, a translate of a definable compact complex manifolds. subgroup. A Cantor-Bendixon analysis of Stone spaces yields a definition of (possibly ordinal-valued or I cannot resist pointing out how the “Mordell- undefined) dimension for definable sets, called Lang conjecture” over function fields in positive Morley rank or Morley dimension. Below we will characteristic proved by Hrushovski (1996) is an refer to this as just dim(). In the context of sets almost immediate consequence of the modularity defined in algebraically closed fields, this coin- of certain definable groups in separably closed cides with algebraic-geometric dimension. But in the fields. There is a slight twist in that we will be con- case of compact complex manifolds it is normally sidering sets defined by a countable collection of only bounded above by the complex dimension. formulas (type-definable sets) rather than by a Shelah’s general theory explains how finite- single formula. But there is no essential differ- dimensional definable sets are built up, by a finite ence in the general theory. sequence of fibrations, from minimal definable So we consider separably closed fields K of sets. The manner in which definable sets interact characteristic p>0 with the feature that K is finite-dimensional of dimension > 1 as a vector is also well understood. Definable sets X1 and X2 are said to be orthogonal if, up to finite Boolean space over its pth powers. We work with the structure (K,+T, ), which we assume to be satu- combination, any definable subset of X1 X2 is of pn rated. Let k = n K . Let A be a simple abelian the form Z1 Z2 for Zi a definable subset of Xi . variety defined over K that does not descend to On minimal definable sets, nonorthogonality is an k (namely, is not rationally isomorphic to an abelian equivalence relation. In fact, minimal definable T variety defined over k). Let A] = (pnA(K)) be the sets X and X will be nonorthogonal if and only n 1 2 group of infinitely p-divisible elements of A(K). if there is a finite-to-finite definable relation Both k and A] are type-definable. The validity of Y X X projecting onto both X and X . 1 2 1 2 the Zilber dichotomy for separably closed fields Among the most important and subtle aspects yields: of geometric stability theory is the distinction between “linear” and “nonlinear” behavior of Lemma 1. A] is minimal and modular. definable sets. What I call linearity is usually The point is that if not, then there is a minimal called “modularity” for lattice-theoretic reasons. field nonorthogonal to A]. It has to be k. The Fix a stable saturated structure M and a finite- nonorthogonality yields a rational isomorphism dimensional definable set X. We call X linear between A and an abelian variety defined over k, (modular) if there are no “large” families of contradiction. It follows that definable subsets of any Xk. By a large such family we mean a definable family of definable Lemma 2. If A is a (not necessarily simple) abelian ] sets Ya for a ∈ Z such that dim(Ya)=n for variety over K with k-trace 0, then A is modular. all a ∈ Z, dim(Z)=m, dim(Y ∩ Y ) dim( ∈ Y ). a Z a for abelian varieties over function fields in posi- If X happens to be minimal, to check linearity tive characteristic. it suffices to consider definable subsets of X X. Intuitively (for X minimal), X is linear if and only Theorem 3. If A is an abelian variety over a if any definable family of minimal subsets of X X function field K/k with k-trace 0, is a finitely is at most a 1-parameter family. Zilber conjectured generated subgroup of A(K), and X is a subvariety a long time ago that a fundamental dichotomy of A defined over K with X ∩ Zariski-dense in X, holds for minimal sets in stable structures. Either then X is a translate of an abelian subvariety of A.

1380 NOTICES OF THE AMS VOLUME 47, NUMBER 11 fea-pillay.qxp 10/16/00 9:03 AM Page 1381

Sketch of proof. Replacing K by its separable clo- sure, we may assume that K and k are as in the previous paragraph; by a transfer argument we can assume (K,+, ) to be saturated. For each n, pn has finite index in , and so some translate of pn meets X in a Zariski-dense set. The same is true of some translate of pnA(K) . Saturation of K implies that after translation of X, A] meets X in a Zariski-dense set. Modularity of A] as in Lemma 2 implies, by Theorem 2, that X ∩ A] is a translate of a subgroup. The same is thus true of X. It was discovered recently that the machinery of stability theory (dimension, orthogonality,…) is valid, but with a few interesting twists, in a much wider setting than stable structures—so-called “simple structures” and their theories. A great deal of energy on the pure side of model theory is currently devoted to understanding this wider class. Although the real and p-adic fields do not fall into this category, finite fields (or rather their limits, pseudofinite fields), the random graph, and sufficiently rich difference fields (fields equipped with an automorphism) do. Making use of defin- ability in difference fields and the validity of the Zilber dichotomy in this context (Chatzidakis- Hrushovski 1999), Hrushovski found another proof of the “Manin-Mumford conjecture” on the inter- section of the torsion points of abelian varieties A defined over number fields with subvarieties of A , obtaining, by virtue of model-theoretic methods, better bounds than previously known. The hard qualitative and quantitative questions concerning rational points of varieties over num- ber fields remain untouched by the model-theoretic methods discussed in this section. Making progress here is a major challenge.

References [1] E. BOUSCAREN, ed., Model Theory and Algebraic Geometry, Lecture Notes in Math., vol. 1696, Springer- Verlag, Berlin, 1998. [2] C. C. CHANG and H. J. KEISLER, Model Theory, 3rd edition, North-Holland, Amsterdam, 1990. [3] L. VAN DEN DRIES, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Notes Ser., vol. 248, Cambridge Univ. Press, Cambridge, 1998. [4] D. HASKELL, A. PILLAY, and C. STEINHORN, eds., Model The- ory, Algebra and Geometry, MSRI Publ., Cambridge Univ. Press, 2000. [5] A. PILLAY, Model theory and Diophantine geometry, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 405–422.

DECEMBER 2000 NOTICES OF THE AMS 1381