Model theory, stability, applications

Anand Pillay

University of Leeds

June 6, 2013

Logic I

I Modern mathematical logic developed at the end of the 19thand beginning of 20th centuries with the so-calledfoundational crisis or crises.

I There was a greater interest in mathematical rigor, and aconcern whether reasoning involving certain infinite quantitieswas sound.

I In addition to logicians such as Cantor, Frege, Russell, majormathematicians of the time such as Hilbert and Poincareparticipated in these developments.

I Out of all of this came the beginnings of mathematicalaccounts of higher level or “metamathematical” notions suchas set, truth, proof, and algorithm (or effective procedure).

Logic I

I Modern mathematical logic developed at the end of the 19thand beginning of 20th centuries with the so-calledfoundational crisis or crises.

I There was a greater interest in mathematical rigor, and aconcern whether reasoning involving certain infinite quantitieswas sound.

I In addition to logicians such as Cantor, Frege, Russell, majormathematicians of the time such as Hilbert and Poincareparticipated in these developments.

I Out of all of this came the beginnings of mathematicalaccounts of higher level or “metamathematical” notions suchas set, truth, proof, and algorithm (or effective procedure).

Logic I

I Modern mathematical logic developed at the end of the 19thand beginning of 20th centuries with the so-calledfoundational crisis or crises.

I There was a greater interest in mathematical rigor, and aconcern whether reasoning involving certain infinite quantitieswas sound.

I In addition to logicians such as Cantor, Frege, Russell, majormathematicians of the time such as Hilbert and Poincareparticipated in these developments.

I Out of all of this came the beginnings of mathematicalaccounts of higher level or “metamathematical” notions suchas set, truth, proof, and algorithm (or effective procedure).

Logic I

I Modern mathematical logic developed at the end of the 19thand beginning of 20th centuries with the so-calledfoundational crisis or crises.

I There was a greater interest in mathematical rigor, and aconcern whether reasoning involving certain infinite quantitieswas sound.

I In addition to logicians such as Cantor, Frege, Russell, majormathematicians of the time such as Hilbert and Poincareparticipated in these developments.

I Out of all of this came the beginnings of mathematicalaccounts of higher level or “metamathematical” notions suchas set, truth, proof, and algorithm (or effective procedure).

Logic II

I These four notions are still at the base of the main areas ofmathematical logic: set theory, model theory, proof theory,and recursion theory, respectively.

I Classical foundational issues are still present in modernmathematical logic, especially set theory.

I But various relations between logic and other areas havedeveloped: set theory has close connections to analysis, prooftheory to computer science, category theory and recentlyhomotopy theory.

I We will discuss in more detail the case of model theory. Earlydevelopments include Malcev’s applications to group theory,Tarski’s analysis of definability in the field of real numbers,and Robinson’s rigorous account of infinitesimals(nonstandard analysis).

Logic II

I These four notions are still at the base of the main areas ofmathematical logic: set theory, model theory, proof theory,and recursion theory, respectively.

I Classical foundational issues are still present in modernmathematical logic, especially set theory.

I But various relations between logic and other areas havedeveloped: set theory has close connections to analysis, prooftheory to computer science, category theory and recentlyhomotopy theory.

I We will discuss in more detail the case of model theory. Earlydevelopments include Malcev’s applications to group theory,Tarski’s analysis of definability in the field of real numbers,and Robinson’s rigorous account of infinitesimals(nonstandard analysis).

Logic II

I These four notions are still at the base of the main areas ofmathematical logic: set theory, model theory, proof theory,and recursion theory, respectively.

I Classical foundational issues are still present in modernmathematical logic, especially set theory.

I But various relations between logic and other areas havedeveloped: set theory has close connections to analysis, prooftheory to computer science, category theory and recentlyhomotopy theory.

I We will discuss in more detail the case of model theory. Earlydevelopments include Malcev’s applications to group theory,Tarski’s analysis of definability in the field of real numbers,and Robinson’s rigorous account of infinitesimals(nonstandard analysis).

Logic II

I These four notions are still at the base of the main areas ofmathematical logic: set theory, model theory, proof theory,and recursion theory, respectively.

I Classical foundational issues are still present in modernmathematical logic, especially set theory.

I But various relations between logic and other areas havedeveloped: set theory has close connections to analysis, prooftheory to computer science, category theory and recentlyhomotopy theory.

I We will discuss in more detail the case of model theory. Earlydevelopments include Malcev’s applications to group theory,Tarski’s analysis of definability in the field of real numbers,and Robinson’s rigorous account of infinitesimals(nonstandard analysis).

Model theory I

I What is model theory?

I It is often thought of as a collection of techniques and notions(compactness, quantifier elimination, o-minimality,..) whichcome to life in applications.

I But there is a “model theory for its own sake” which I wouldtentatively define as the classification of first order theories.

I A first order theory T is at the naive level simply a collectionof “first order sentences” in some vocabulary L with relation,function and constant symbols as well as the usual logicalconnectives “and”, “or”, “not”, and quantifiers “there exist”,“for all”.

I “First order” refers to the quantifiers ranging over elements orindividuals rather than sets.

Model theory I

I What is model theory?

I It is often thought of as a collection of techniques and notions(compactness, quantifier elimination, o-minimality,..) whichcome to life in applications.

I But there is a “model theory for its own sake” which I wouldtentatively define as the classification of first order theories.

I A first order theory T is at the naive level simply a collectionof “first order sentences” in some vocabulary L with relation,function and constant symbols as well as the usual logicalconnectives “and”, “or”, “not”, and quantifiers “there exist”,“for all”.

I “First order” refers to the quantifiers ranging over elements orindividuals rather than sets.

Model theory I

I What is model theory?

I It is often thought of as a collection of techniques and notions(compactness, quantifier elimination, o-minimality,..) whichcome to life in applications.

I But there is a “model theory for its own sake” which I wouldtentatively define as the classification of first order theories.

I A first order theory T is at the naive level simply a collectionof “first order sentences” in some vocabulary L with relation,function and constant symbols as well as the usual logicalconnectives “and”, “or”, “not”, and quantifiers “there exist”,“for all”.

I “First order” refers to the quantifiers ranging over elements orindividuals rather than sets.

Model theory I

I What is model theory?

I It is often thought of as a collection of techniques and notions(compactness, quantifier elimination, o-minimality,..) whichcome to life in applications.

I But there is a “model theory for its own sake” which I wouldtentatively define as the classification of first order theories.

I A first order theory T is at the naive level simply a collectionof “first order sentences” in some vocabulary L with relation,function and constant symbols as well as the usual logicalconnectives “and”, “or”, “not”, and quantifiers “there exist”,“for all”.

I “First order” refers to the quantifiers ranging over elements orindividuals rather than sets.

Model theory I

I What is model theory?

I It is often thought of as a collection of techniques and notions(compactness, quantifier elimination, o-minimality,..) whichcome to life in applications.

I But there is a “model theory for its own sake” which I wouldtentatively define as the classification of first order theories.

I A first order theory T is at the naive level simply a collectionof “first order sentences” in some vocabulary L with relation,function and constant symbols as well as the usual logicalconnectives “and”, “or”, “not”, and quantifiers “there exist”,“for all”.

I “First order” refers to the quantifiers ranging over elements orindividuals rather than sets.

Model theory II

I A model of T is simply a first order structure M consisting ofan underlying set or universe M together with a distinguishedcollection of relations (subsets of Mn), functions Mn →Mand “constants” corresponding to the symbols of L, in whichthe sentences of T are true. It is natural to allow severaluniverses (many-sorted framework).

I There is a tautological aspect here: the set of axioms forgroups is a first order theory in an appropriate language, and amodel of T is just a group.

I On the other hand, the axioms for topological spaces, andtopological spaces themselves have on the face of it a “secondorder” character. (A set X is given the structure of atopological space by specifying a collection of subsets of Xsatisfying various properties..).

Model theory II

I A model of T is simply a first order structure M consisting ofan underlying set or universe M together with a distinguishedcollection of relations (subsets of Mn), functions Mn →Mand “constants” corresponding to the symbols of L, in whichthe sentences of T are true. It is natural to allow severaluniverses (many-sorted framework).

I There is a tautological aspect here: the set of axioms forgroups is a first order theory in an appropriate language, and amodel of T is just a group.

I On the other hand, the axioms for topological spaces, andtopological spaces themselves have on the face of it a “secondorder” character. (A set X is given the structure of atopological space by specifying a collection of subsets of Xsatisfying various properties..).

Model theory II

I A model of T is simply a first order structure M consisting ofan underlying set or universe M together with a distinguishedcollection of relations (subsets of Mn), functions Mn →Mand “constants” corresponding to the symbols of L, in whichthe sentences of T are true. It is natural to allow severaluniverses (many-sorted framework).

I There is a tautological aspect here: the set of axioms forgroups is a first order theory in an appropriate language, and amodel of T is just a group.

I On the other hand, the axioms for topological spaces, andtopological spaces themselves have on the face of it a “secondorder” character. (A set X is given the structure of atopological space by specifying a collection of subsets of Xsatisfying various properties..).

Definable sets I

I Another key notion is that of a definable set.

I If (G, ·) is a group, and a ∈ G then the collection of elementsof G which commute with a is the solution set of an“equation”, x · a = a · x.

I However Z(G), the centre of G, which is the collection ofelements of G which commute with every element of G, is“defined by” the first order formula ∀y(x · y = y · x).

I In the structure (R,+, ·,−) the ordering x ≤ y is defined bythe first order formula ∃z(y − x = z2).

I Our familiar number systems already provide quite differentbehaviour or features of definable sets.

Definable sets I

I Another key notion is that of a definable set.

I If (G, ·) is a group, and a ∈ G then the collection of elementsof G which commute with a is the solution set of an“equation”, x · a = a · x.

I However Z(G), the centre of G, which is the collection ofelements of G which commute with every element of G, is“defined by” the first order formula ∀y(x · y = y · x).

I In the structure (R,+, ·,−) the ordering x ≤ y is defined bythe first order formula ∃z(y − x = z2).

I Our familiar number systems already provide quite differentbehaviour or features of definable sets.

Definable sets I

I Another key notion is that of a definable set.

I If (G, ·) is a group, and a ∈ G then the collection of elementsof G which commute with a is the solution set of an“equation”, x · a = a · x.

I However Z(G), the centre of G, which is the collection ofelements of G which commute with every element of G, is“defined by” the first order formula ∀y(x · y = y · x).

I In the structure (R,+, ·,−) the ordering x ≤ y is defined bythe first order formula ∃z(y − x = z2).

I Our familiar number systems already provide quite differentbehaviour or features of definable sets.

Definable sets I

I Another key notion is that of a definable set.

I If (G, ·) is a group, and a ∈ G then the collection of elementsof G which commute with a is the solution set of an“equation”, x · a = a · x.

I However Z(G), the centre of G, which is the collection ofelements of G which commute with every element of G, is“defined by” the first order formula ∀y(x · y = y · x).

I In the structure (R,+, ·,−) the ordering x ≤ y is defined bythe first order formula ∃z(y − x = z2).

I Our familiar number systems already provide quite differentbehaviour or features of definable sets.

Definable sets II

I In the structure (N,+,×, 0, 1), subsets of N definable byformulas φ(x) which begin with a sequence of quantifiers∃y1∀y2∃y3...∀yn get more complicated as n increases.

I The collection of definable subsets of N is called thearithmetical hierarchy, and already with one existentialquantifier we can define “noncomputable” sets. In fact thestudy of definability in (N,+,×, 0, 1) is precisely recursiontheory.

I However in the structure (R,+, ·), the hierarchy collapses, oneonly needs one block of existential quantifiers to definedefinable sets. Moreover the definable sets have a geometricfeature: they are the so-called semialgebraic sets, namelyfinite unions of subsets of Rn of form{x : f(x) = 0 ∧

∧i=1,..k gi(x) > 0} where f and the gi are

polynomials with coefficients from R.

Definable sets II

I In the structure (N,+,×, 0, 1), subsets of N definable byformulas φ(x) which begin with a sequence of quantifiers∃y1∀y2∃y3...∀yn get more complicated as n increases.

I The collection of definable subsets of N is called thearithmetical hierarchy, and already with one existentialquantifier we can define “noncomputable” sets. In fact thestudy of definability in (N,+,×, 0, 1) is precisely recursiontheory.

I However in the structure (R,+, ·), the hierarchy collapses, oneonly needs one block of existential quantifiers to definedefinable sets. Moreover the definable sets have a geometricfeature: they are the so-called semialgebraic sets, namelyfinite unions of subsets of Rn of form{x : f(x) = 0 ∧

∧i=1,..k gi(x) > 0} where f and the gi are

polynomials with coefficients from R.

Definable sets II

I In the structure (N,+,×, 0, 1), subsets of N definable byformulas φ(x) which begin with a sequence of quantifiers∃y1∀y2∃y3...∀yn get more complicated as n increases.

I The collection of definable subsets of N is called thearithmetical hierarchy, and already with one existentialquantifier we can define “noncomputable” sets. In fact thestudy of definability in (N,+,×, 0, 1) is precisely recursiontheory.

I However in the structure (R,+, ·), the hierarchy collapses, oneonly needs one block of existential quantifiers to definedefinable sets. Moreover the definable sets have a geometricfeature: they are the so-called semialgebraic sets, namelyfinite unions of subsets of Rn of form{x : f(x) = 0 ∧

∧i=1,..k gi(x) > 0} where f and the gi are

polynomials with coefficients from R.

Definable sets III

I In the case of the structure (C,+,×) it is even better: thehierarchy collapses to sets defined without any quantifiers :the definable sets are precisely the constructible sets: finiteBoolean combinations of algebraic varieties. (Chevalley’stheorem.)

I The model-theoretic problem of describing definable sets in(C,+,×) up to definable bijection is essentially the same asthe central problem in algebraic geometry, namelyclassification of algebraic varieties up to birationalisomorphism.

I But the model theory of the structure (C,+,×) or its firstorder theory ACF0, has little bearing on the problem, and it israther definability in richer (but still tame) structures such asfields equipped with a derivation, valuation, automorphism...which has consequences and applications.

Definable sets III

I In the case of the structure (C,+,×) it is even better: thehierarchy collapses to sets defined without any quantifiers :the definable sets are precisely the constructible sets: finiteBoolean combinations of algebraic varieties. (Chevalley’stheorem.)

I The model-theoretic problem of describing definable sets in(C,+,×) up to definable bijection is essentially the same asthe central problem in algebraic geometry, namelyclassification of algebraic varieties up to birationalisomorphism.

I But the model theory of the structure (C,+,×) or its firstorder theory ACF0, has little bearing on the problem, and it israther definability in richer (but still tame) structures such asfields equipped with a derivation, valuation, automorphism...which has consequences and applications.

Definable sets III

I In the case of the structure (C,+,×) it is even better: thehierarchy collapses to sets defined without any quantifiers :the definable sets are precisely the constructible sets: finiteBoolean combinations of algebraic varieties. (Chevalley’stheorem.)

I The model-theoretic problem of describing definable sets in(C,+,×) up to definable bijection is essentially the same asthe central problem in algebraic geometry, namelyclassification of algebraic varieties up to birationalisomorphism.

I But the model theory of the structure (C,+,×) or its firstorder theory ACF0, has little bearing on the problem, and it israther definability in richer (but still tame) structures such asfields equipped with a derivation, valuation, automorphism...which has consequences and applications.

First order theories

I We will restrict our attention to complete theories T , namelytheories which decide every sentence of the vocabulary. Forexample ACF0, the axioms for algebraically closed fields ofcharacteristic 0.

I Attached to a first order theory there are at least twocategories, Mod(T ) the category of models of T , andDef(T ) the category of definable sets, where the latter canbe identified with Def(M), the category of definable sets in a“big” model M of T .

I The classification of first order theories concerns findingmeaningful dividing lines. The “logically perfect” first ordertheories are the stable theories, to be discussed below.

First order theories

I We will restrict our attention to complete theories T , namelytheories which decide every sentence of the vocabulary. Forexample ACF0, the axioms for algebraically closed fields ofcharacteristic 0.

I Attached to a first order theory there are at least twocategories, Mod(T ) the category of models of T , andDef(T ) the category of definable sets, where the latter canbe identified with Def(M), the category of definable sets in a“big” model M of T .

I The classification of first order theories concerns findingmeaningful dividing lines. The “logically perfect” first ordertheories are the stable theories, to be discussed below.

First order theories

I We will restrict our attention to complete theories T , namelytheories which decide every sentence of the vocabulary. Forexample ACF0, the axioms for algebraically closed fields ofcharacteristic 0.

I Attached to a first order theory there are at least twocategories, Mod(T ) the category of models of T , andDef(T ) the category of definable sets, where the latter canbe identified with Def(M), the category of definable sets in a“big” model M of T .

I The classification of first order theories concerns findingmeaningful dividing lines. The “logically perfect” first ordertheories are the stable theories, to be discussed below.

Stability I

I Model theory in the 1960’s and 70’s had a very“set-theoretic” character (influenced by Tarski among others)and the original questions which led to the development of thesubject as something for its own sake have this form.

I For example the spectrum problem: given a (complete)theory, we have the function I(−T ) from (infinite) cardinalsto cardinals, where I(κ, T ) is the number of models of T ofcardinality κ, up to isomorphism. What are the possible suchfunctions, as T varies?

I Shelah solved the problem for countable theories, in theprocess identifying the class of stable first order theories, anddeveloping stability theory, the detailed analysis of thecategories Mod(T ) and Def(T ) for an arbitrary stable theoryT .

Stability I

I Model theory in the 1960’s and 70’s had a very“set-theoretic” character (influenced by Tarski among others)and the original questions which led to the development of thesubject as something for its own sake have this form.

I For example the spectrum problem: given a (complete)theory, we have the function I(−T ) from (infinite) cardinalsto cardinals, where I(κ, T ) is the number of models of T ofcardinality κ, up to isomorphism. What are the possible suchfunctions, as T varies?

I Shelah solved the problem for countable theories, in theprocess identifying the class of stable first order theories, anddeveloping stability theory, the detailed analysis of thecategories Mod(T ) and Def(T ) for an arbitrary stable theoryT .

Stability I

I Model theory in the 1960’s and 70’s had a very“set-theoretic” character (influenced by Tarski among others)and the original questions which led to the development of thesubject as something for its own sake have this form.

I For example the spectrum problem: given a (complete)theory, we have the function I(−T ) from (infinite) cardinalsto cardinals, where I(κ, T ) is the number of models of T ofcardinality κ, up to isomorphism. What are the possible suchfunctions, as T varies?

I Shelah solved the problem for countable theories, in theprocess identifying the class of stable first order theories, anddeveloping stability theory, the detailed analysis of thecategories Mod(T ) and Def(T ) for an arbitrary stable theoryT .

Stability II

I So although the spectrum problem is about Mod(T ), Shelah’swork gave an enormous amount of information about andtools for understanding Def(T ) (when T is stable).

I The (or a) definition of stability is not particularly enlighteningbut is a good example of a “model-theoretic” property: T isstable if there is no model M of T definable relation R(x, y)and ai, bi ∈M for i = 1, 2, .. such that R(ai, bj) if i < j.

I ACF0 is the canonical example of a stable theory. Another(complete) example is the theory of infinite vector spaces overa fixed division ring.

I More recently it was discovered (Sela) that the first ordertheory of the free group (F2, ·) is stable, yielding newconnections between model theory and geometric grouptheory.

Stability II

I So although the spectrum problem is about Mod(T ), Shelah’swork gave an enormous amount of information about andtools for understanding Def(T ) (when T is stable).

I The (or a) definition of stability is not particularly enlighteningbut is a good example of a “model-theoretic” property: T isstable if there is no model M of T definable relation R(x, y)and ai, bi ∈M for i = 1, 2, .. such that R(ai, bj) if i < j.

I ACF0 is the canonical example of a stable theory. Another(complete) example is the theory of infinite vector spaces overa fixed division ring.

I More recently it was discovered (Sela) that the first ordertheory of the free group (F2, ·) is stable, yielding newconnections between model theory and geometric grouptheory.

Stability II

I So although the spectrum problem is about Mod(T ), Shelah’swork gave an enormous amount of information about andtools for understanding Def(T ) (when T is stable).

I The (or a) definition of stability is not particularly enlighteningbut is a good example of a “model-theoretic” property: T isstable if there is no model M of T definable relation R(x, y)and ai, bi ∈M for i = 1, 2, .. such that R(ai, bj) if i < j.

I ACF0 is the canonical example of a stable theory. Another(complete) example is the theory of infinite vector spaces overa fixed division ring.

I More recently it was discovered (Sela) that the first ordertheory of the free group (F2, ·) is stable, yielding newconnections between model theory and geometric grouptheory.

Stability II

I So although the spectrum problem is about Mod(T ), Shelah’swork gave an enormous amount of information about andtools for understanding Def(T ) (when T is stable).

I The (or a) definition of stability is not particularly enlighteningbut is a good example of a “model-theoretic” property: T isstable if there is no model M of T definable relation R(x, y)and ai, bi ∈M for i = 1, 2, .. such that R(ai, bj) if i < j.

I ACF0 is the canonical example of a stable theory. Another(complete) example is the theory of infinite vector spaces overa fixed division ring.

I More recently it was discovered (Sela) that the first ordertheory of the free group (F2, ·) is stable, yielding newconnections between model theory and geometric grouptheory.

Finite rank stable theories I

I Among the more tractable classes of stable theories are thoseof “finite rank”, i.e. where all definable sets X have finiterank/dimension, in a sense that we describe now:

I The relevant dimension notion is traditionally called “Morleyrank” and is simply Cantor-Bendixon rank on the Booleanalgebra of definable (with parameters, in an ambient saturatedmodel) subsets of X:

I X has Morley rank 0 if X is finite, in which case themultiplicity of X is its cardinality.

I X has Morley rank n+ 1 and multiplicity 1 if it has Morleyrank > n and cannot be partitioned into two definable subsetsof rank > n.

I The building blocks of all definable sets in a finite rank stabletheory (in a sense that I will say something about if there istime) are what I will call the minimal definable sets.

Finite rank stable theories I

I Among the more tractable classes of stable theories are thoseof “finite rank”, i.e. where all definable sets X have finiterank/dimension, in a sense that we describe now:

I The relevant dimension notion is traditionally called “Morleyrank” and is simply Cantor-Bendixon rank on the Booleanalgebra of definable (with parameters, in an ambient saturatedmodel) subsets of X:

I X has Morley rank 0 if X is finite, in which case themultiplicity of X is its cardinality.

I X has Morley rank n+ 1 and multiplicity 1 if it has Morleyrank > n and cannot be partitioned into two definable subsetsof rank > n.

I The building blocks of all definable sets in a finite rank stabletheory (in a sense that I will say something about if there istime) are what I will call the minimal definable sets.

Finite rank stable theories I

I Among the more tractable classes of stable theories are thoseof “finite rank”, i.e. where all definable sets X have finiterank/dimension, in a sense that we describe now:

I The relevant dimension notion is traditionally called “Morleyrank” and is simply Cantor-Bendixon rank on the Booleanalgebra of definable (with parameters, in an ambient saturatedmodel) subsets of X:

I X has Morley rank 0 if X is finite, in which case themultiplicity of X is its cardinality.

I X has Morley rank n+ 1 and multiplicity 1 if it has Morleyrank > n and cannot be partitioned into two definable subsetsof rank > n.

I The building blocks of all definable sets in a finite rank stabletheory (in a sense that I will say something about if there istime) are what I will call the minimal definable sets.

Finite rank stable theories I

I Among the more tractable classes of stable theories are thoseof “finite rank”, i.e. where all definable sets X have finiterank/dimension, in a sense that we describe now:

I The relevant dimension notion is traditionally called “Morleyrank” and is simply Cantor-Bendixon rank on the Booleanalgebra of definable (with parameters, in an ambient saturatedmodel) subsets of X:

I X has Morley rank 0 if X is finite, in which case themultiplicity of X is its cardinality.

I X has Morley rank n+ 1 and multiplicity 1 if it has Morleyrank > n and cannot be partitioned into two definable subsetsof rank > n.

I The building blocks of all definable sets in a finite rank stabletheory (in a sense that I will say something about if there istime) are what I will call the minimal definable sets.

Finite rank stable theories I

I Among the more tractable classes of stable theories are thoseof “finite rank”, i.e. where all definable sets X have finiterank/dimension, in a sense that we describe now:

I The relevant dimension notion is traditionally called “Morleyrank” and is simply Cantor-Bendixon rank on the Booleanalgebra of definable (with parameters, in an ambient saturatedmodel) subsets of X:

I X has Morley rank 0 if X is finite, in which case themultiplicity of X is its cardinality.

I X has Morley rank n+ 1 and multiplicity 1 if it has Morleyrank > n and cannot be partitioned into two definable subsetsof rank > n.

I The building blocks of all definable sets in a finite rank stabletheory (in a sense that I will say something about if there istime) are what I will call the minimal definable sets.

Finite rank stable theories II

I Loosely speaking X is minimal if X is infinite and“generically” it cannot be partitioned into 2 infinite definablesets.

I A special case is strongly minimal meaning precisely Morleyrank and multiplicity 1, namely X cannot be partitioned into2 infinite definable sets.

I There is a natural equivalence relation on minimal sets:X ∼ Y if there is a definable Z ⊆ X × Y projectinggenerically finite-to-one on each of X,Y (a definableself-correspondence).

I A very influential conjecture of Boris Zilber was that that anyminimal set (in a finite rank stable theory) is of three possible(mutually exclusive) types:

I (a) “field like”: up to ∼, X has definably the structure of analgebraically closed field

Finite rank stable theories II

I Loosely speaking X is minimal if X is infinite and“generically” it cannot be partitioned into 2 infinite definablesets.

I A special case is strongly minimal meaning precisely Morleyrank and multiplicity 1, namely X cannot be partitioned into2 infinite definable sets.

I There is a natural equivalence relation on minimal sets:X ∼ Y if there is a definable Z ⊆ X × Y projectinggenerically finite-to-one on each of X,Y (a definableself-correspondence).

I A very influential conjecture of Boris Zilber was that that anyminimal set (in a finite rank stable theory) is of three possible(mutually exclusive) types:

I (a) “field like”: up to ∼, X has definably the structure of analgebraically closed field

Finite rank stable theories II

I Loosely speaking X is minimal if X is infinite and“generically” it cannot be partitioned into 2 infinite definablesets.

I A special case is strongly minimal meaning precisely Morleyrank and multiplicity 1, namely X cannot be partitioned into2 infinite definable sets.

I There is a natural equivalence relation on minimal sets:X ∼ Y if there is a definable Z ⊆ X × Y projectinggenerically finite-to-one on each of X,Y (a definableself-correspondence).

I A very influential conjecture of Boris Zilber was that that anyminimal set (in a finite rank stable theory) is of three possible(mutually exclusive) types:

I (a) “field like”: up to ∼, X has definably the structure of analgebraically closed field

Finite rank stable theories II

I Loosely speaking X is minimal if X is infinite and“generically” it cannot be partitioned into 2 infinite definablesets.

I A special case is strongly minimal meaning precisely Morleyrank and multiplicity 1, namely X cannot be partitioned into2 infinite definable sets.

I There is a natural equivalence relation on minimal sets:X ∼ Y if there is a definable Z ⊆ X × Y projectinggenerically finite-to-one on each of X,Y (a definableself-correspondence).

I A very influential conjecture of Boris Zilber was that that anyminimal set (in a finite rank stable theory) is of three possible(mutually exclusive) types:

I (a) “field like”: up to ∼, X has definably the structure of analgebraically closed field

Finite rank stable theories II

I Loosely speaking X is minimal if X is infinite and“generically” it cannot be partitioned into 2 infinite definablesets.

I A special case is strongly minimal meaning precisely Morleyrank and multiplicity 1, namely X cannot be partitioned into2 infinite definable sets.

I There is a natural equivalence relation on minimal sets:X ∼ Y if there is a definable Z ⊆ X × Y projectinggenerically finite-to-one on each of X,Y (a definableself-correspondence).

I A very influential conjecture of Boris Zilber was that that anyminimal set (in a finite rank stable theory) is of three possible(mutually exclusive) types:

I (a) “field like”: up to ∼, X has definably the structure of analgebraically closed field

Finite rank stable theories III

I (b) ”vector space like”: up to ∼, X has a definablecommutative group structure such that moreover anydefinable subset of X × ..X is up to finite Booleancombination and translation, a definable subgroup.

I (c) X is “trivial”: there is no infinite definable family ofdefinable self correspondences of X.

I A counterexample was found by Hrushovski in the late 80’s,and the methods for constructing such examples have becomea subarea of model theory.

I However the conjecture has been proved for some very richfinite rank stable theories (originally via so-called Zariskigeometries, but other proofs were found later), and in the lastpart of the talk (if there is time) I will discuss a couple ofexamples and applications with algebraic-geometric andnumber-theoretic features.

Finite rank stable theories III

I (b) ”vector space like”: up to ∼, X has a definablecommutative group structure such that moreover anydefinable subset of X × ..X is up to finite Booleancombination and translation, a definable subgroup.

I (c) X is “trivial”: there is no infinite definable family ofdefinable self correspondences of X.

I A counterexample was found by Hrushovski in the late 80’s,and the methods for constructing such examples have becomea subarea of model theory.

I However the conjecture has been proved for some very richfinite rank stable theories (originally via so-called Zariskigeometries, but other proofs were found later), and in the lastpart of the talk (if there is time) I will discuss a couple ofexamples and applications with algebraic-geometric andnumber-theoretic features.

Finite rank stable theories III

I (b) ”vector space like”: up to ∼, X has a definablecommutative group structure such that moreover anydefinable subset of X × ..X is up to finite Booleancombination and translation, a definable subgroup.

I (c) X is “trivial”: there is no infinite definable family ofdefinable self correspondences of X.

I A counterexample was found by Hrushovski in the late 80’s,and the methods for constructing such examples have becomea subarea of model theory.

I However the conjecture has been proved for some very richfinite rank stable theories (originally via so-called Zariskigeometries, but other proofs were found later), and in the lastpart of the talk (if there is time) I will discuss a couple ofexamples and applications with algebraic-geometric andnumber-theoretic features.

Finite rank stable theories III

I (b) ”vector space like”: up to ∼, X has a definablecommutative group structure such that moreover anydefinable subset of X × ..X is up to finite Booleancombination and translation, a definable subgroup.

I (c) X is “trivial”: there is no infinite definable family ofdefinable self correspondences of X.

I A counterexample was found by Hrushovski in the late 80’s,and the methods for constructing such examples have becomea subarea of model theory.

I However the conjecture has been proved for some very richfinite rank stable theories (originally via so-called Zariskigeometries, but other proofs were found later), and in the lastpart of the talk (if there is time) I will discuss a couple ofexamples and applications with algebraic-geometric andnumber-theoretic features.

DCF0 I

I DCF0 is the theory of differentially closed fields ofcharacteristic 0, the theory of a “universal” differential field(U ,+,×, ∂).

I DCF0 is stable and of infinite rank, but the family of finiterank definable sets can be considered as a many-sorted sortedtheory or structure.

I Algebraic geometry (ACF0) lives on the field of constants C.

I The Zilber conjecture is valid in this context (and gave rise tonew results in diophantine geometry over function fields):minimal sets of type (a) are algebraic curves in C, minimalsets of type (b) are related to simple “nonconstant” abelianvarieties, and there is an interest in identifying minimal sets oftype (c).

DCF0 I

I DCF0 is the theory of differentially closed fields ofcharacteristic 0, the theory of a “universal” differential field(U ,+,×, ∂).

I DCF0 is stable and of infinite rank, but the family of finiterank definable sets can be considered as a many-sorted sortedtheory or structure.

I Algebraic geometry (ACF0) lives on the field of constants C.

I The Zilber conjecture is valid in this context (and gave rise tonew results in diophantine geometry over function fields):minimal sets of type (a) are algebraic curves in C, minimalsets of type (b) are related to simple “nonconstant” abelianvarieties, and there is an interest in identifying minimal sets oftype (c).

DCF0 I

I DCF0 is the theory of differentially closed fields ofcharacteristic 0, the theory of a “universal” differential field(U ,+,×, ∂).

I DCF0 is stable and of infinite rank, but the family of finiterank definable sets can be considered as a many-sorted sortedtheory or structure.

I Algebraic geometry (ACF0) lives on the field of constants C.

I The Zilber conjecture is valid in this context (and gave rise tonew results in diophantine geometry over function fields):minimal sets of type (a) are algebraic curves in C, minimalsets of type (b) are related to simple “nonconstant” abelianvarieties, and there is an interest in identifying minimal sets oftype (c).

DCF0 I

I DCF0 is the theory of differentially closed fields ofcharacteristic 0, the theory of a “universal” differential field(U ,+,×, ∂).

I DCF0 is stable and of infinite rank, but the family of finiterank definable sets can be considered as a many-sorted sortedtheory or structure.

I Algebraic geometry (ACF0) lives on the field of constants C.

I The Zilber conjecture is valid in this context (and gave rise tonew results in diophantine geometry over function fields):minimal sets of type (a) are algebraic curves in C, minimalsets of type (b) are related to simple “nonconstant” abelianvarieties, and there is an interest in identifying minimal sets oftype (c).

DCF0 II

I A recent application (with J, Nagloo) is to transcendence(algebraic independence) questions regarding an intensivelystudied class of ordinary differential equations, in the complexdomain, namely the Painleve equations.

Theorem 0.1Consider the Painleve II family of second order ODE’s:y′′ = 2y3 + ty + α where α ∈ C. Then the solution set Yα of therelevant equation (as a definable set in U) is strongly minimal iffα /∈ Z+ 1/2, and moreover for all such α, Yα is of type (c) (trivial).Moreover any “generic” equation in each of the Painleve familiesI-VI, is strongly minimal and “strongly trivial” implying that ify1, .., yn are distinct solutions, then y1, y

′1, ..., yn, y

′n are

algebraically independent over C(t).

CCM

I CCM is the many sorted structure of compact complexmanifolds, where the distinguished relations (on finiteCartesian products of manifolds) are the analytic subvarieties.It is a finite rank stable structure (theory).

I The Zilber conjecture is valid in CCM .

I Algebraic geometry lives on the sort P 1(C), and minimal setsof kind (a) are algebraic curves.

I Some time ago, with Scanlon, we showed that stronglyminimal compact complex manifolds of type (b) are(nonalgebraic simple) complex tori.

I Minimal sets of kind (c) are quite rare, and it is conjecturedthat in the Kahler context they are closely related tohyperkaeler manifolds, another intensively studied class ofcompact complex manifolds.

CCM

I CCM is the many sorted structure of compact complexmanifolds, where the distinguished relations (on finiteCartesian products of manifolds) are the analytic subvarieties.It is a finite rank stable structure (theory).

I The Zilber conjecture is valid in CCM .

I Algebraic geometry lives on the sort P 1(C), and minimal setsof kind (a) are algebraic curves.

I Some time ago, with Scanlon, we showed that stronglyminimal compact complex manifolds of type (b) are(nonalgebraic simple) complex tori.

I Minimal sets of kind (c) are quite rare, and it is conjecturedthat in the Kahler context they are closely related tohyperkaeler manifolds, another intensively studied class ofcompact complex manifolds.

CCM

I CCM is the many sorted structure of compact complexmanifolds, where the distinguished relations (on finiteCartesian products of manifolds) are the analytic subvarieties.It is a finite rank stable structure (theory).

I The Zilber conjecture is valid in CCM .

I Algebraic geometry lives on the sort P 1(C), and minimal setsof kind (a) are algebraic curves.

I Some time ago, with Scanlon, we showed that stronglyminimal compact complex manifolds of type (b) are(nonalgebraic simple) complex tori.

I Minimal sets of kind (c) are quite rare, and it is conjecturedthat in the Kahler context they are closely related tohyperkaeler manifolds, another intensively studied class ofcompact complex manifolds.

CCM

I CCM is the many sorted structure of compact complexmanifolds, where the distinguished relations (on finiteCartesian products of manifolds) are the analytic subvarieties.It is a finite rank stable structure (theory).

I The Zilber conjecture is valid in CCM .

I Algebraic geometry lives on the sort P 1(C), and minimal setsof kind (a) are algebraic curves.

I Some time ago, with Scanlon, we showed that stronglyminimal compact complex manifolds of type (b) are(nonalgebraic simple) complex tori.

I Minimal sets of kind (c) are quite rare, and it is conjecturedthat in the Kahler context they are closely related tohyperkaeler manifolds, another intensively studied class ofcompact complex manifolds.

CCM

I CCM is the many sorted structure of compact complexmanifolds, where the distinguished relations (on finiteCartesian products of manifolds) are the analytic subvarieties.It is a finite rank stable structure (theory).

I The Zilber conjecture is valid in CCM .

I Algebraic geometry lives on the sort P 1(C), and minimal setsof kind (a) are algebraic curves.

I Some time ago, with Scanlon, we showed that stronglyminimal compact complex manifolds of type (b) are(nonalgebraic simple) complex tori.

I Minimal sets of kind (c) are quite rare, and it is conjecturedthat in the Kahler context they are closely related tohyperkaeler manifolds, another intensively studied class ofcompact complex manifolds.