Imaginaries in Model Theory

Martin Hils

Introduction Imaginaries in Model Theory

Imaginary Galois theory Algebraic closure in Meq Martin Hils Elimination of imaginaries Geometric Équipe de Logique Mathématique, Université Paris 7 stability Trichotomy Conjecture The group Philosophy and Model Theory Conference conﬁguration Hyperimagi- Université Paris Ouest & École normale supérieure naries June 2-5, 2010, Paris Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Outline

Imaginaries in Model 1 Introduction Theory

Martin Hils 2 Imaginary Galois theory Algebraic closure in Meq Introduction Elimination of imaginaries Imaginary Galois theory Algebraic 3 Geometric stability closure in Meq Trichotomy Conjecture Elimination of imaginaries The group conﬁguration Geometric stability Trichotomy 4 Hyperimaginaries Conjecture The group Utility in the unstable case conﬁguration Hyperimagi- Positive Logic naries Utility in the 5 Conclusion unstable case Positive Logic What we have learnt Conclusion What we Left outs have learnt Left outs Context

Imaginaries in Model Theory

Martin Hils

Introduction L is some countable ﬁrst order language Imaginary Galois theory (possibly many-sorted); Algebraic closure in Meq T a complete L-theory; Elimination of imaginaries U |= T is very saturated and homogeneous; Geometric stability all models M we consider (and all parameter sets A) are Trichotomy Conjecture The group small, with M 4 U. conﬁguration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Imaginary Elements

Imaginaries in Model Theory

Martin Hils Recall:

Introduction An equivalence relation E on a set D is a binary relation Imaginary which is reflexive, symmetric and transitive; Galois theory Algebraic closure in D is partitioned into the equivalence classes modulo E, Meq 0 0 Elimination of i.e. sets of the form d/E := {d ∈ D | dEd }. imaginaries Geometric stability Definition Trichotomy Conjecture The group An imaginary element in U is an equivalence class d/E, where configuration E is a definable equivalence relation on a definable set D ⊆ Un Hyperimagi- naries and d ∈ D(U). Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Examples of Imaginaries I

Imaginaries in Model Theory Example (Unordered tuples) Martin Hils

Introduction In any theory, the formula

Imaginary Galois theory (x = x0 ∧ y = y 0) ∨ (x = y 0 ∧ y = x0) Algebraic closure in Meq Elimination of 0 0 imaginaries deﬁnes an equiv. relation (x, y)E2(x , y ) on pairs, with Geometric stability 0 0 0 0 Trichotomy (a, b)E2(a , b ) ⇔ {a, b} = {a , b }. Conjecture The group conﬁguration Thus, {a, b} may be thought of as an imaginary element. Hyperimagi- naries Similarly, for any n ∈ , the set {a1,..., an} may be Utility in the N unstable case Positive Logic thought of as an imaginary. Conclusion What we have learnt Left outs Examples of Imaginaries II

Imaginaries in Model Theory

Martin Hils k A group (G, ·) is a definable group in U if G ⊆def U and 3 3k Introduction Γ = {(f , g, h) ∈ G | f · g = h} ⊆def U for some k ∈ N. Imaginary Galois theory Algebraic Example (Cosets) closure in Meq Elimination of Let (G, ·) be definable group in U and H a definable subgroup imaginaries Geometric of G. Then any coset stability Trichotomy Conjecture g · H = {g · h | h ∈ H} The group configuration Hyperimagi- 0 0 naries is an imaginary (w.r.t. gEg ⇔ ∃h ∈ H g · h = g ). Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Examples of Imaginaries III

Imaginaries in Model Theory

Martin Hils Example (Vectors in Aﬃne Space)

Introduction Consider the aﬃne space associated to the Q-vector Imaginary n n Galois theory space Q , i.e. the structure M = hQ , αi, where Algebraic closure in Meq Elimination of α(a, b, c) := a + (c − b). imaginaries Geometric stability The vector bc~ is an imaginary (b, c)/E in M, for Trichotomy Conjecture The group 0 0 0 0 conﬁguration (b, c)E(b , c ):⇔ α(b, b, c) = α(b, b , c ). Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Utility of Imaginaries

Imaginaries in Model Theory Martin Hils Taking into account imaginary elements has several advantages: Introduction

Imaginary Galois theory may talk about quotient objects Algebraic closure in Meq (e.g. G/H, where H ≤ G are definable groups) Elimination of imaginaries ⇒ category of def. objects is closed under quotients; Geometric stability Trichotomy right framework for interpretations; Conjecture The group configuration existence of codes for definable sets Hyperimagi- (will be made precise later). naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Adding Imaginaries: Shelah’s Meq-Construction

Imaginaries in Model Theory

Martin Hils There is a canonical way of adding all imaginaries to M, due to Introduction Shelah, by expanding Imaginary Galois theory Algebraic eq closure in L to a many-sorted language L , Meq Elimination of eq eq imaginaries T to a (complete) L -theory T and Geometric eq eq stability M |= T to M |= T such that Trichotomy eq Conjecture M 7→ M is an equivalence of categories between The group eq conﬁguration hMod(T ), 4i and hMod(T ), 4i. Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Shelah’s Meq-Construction (continued)

Imaginaries in Model Theory For any ∅-definable equivalence relation E on Mn we add Martin Hils a new imaginary sort SE Introduction (the intitial sort of M is called the real sort Sreal ), Imaginary n Galois theory a new function symbol πE : Sreal → SE Algebraic eq closure in ⇒ obtain L ; Meq Elimination of imaginaries axioms stating that πE is surjective, with Geometric stability 0 0 Trichotomy π (a) = π (a ) ⇔ aEa Conjecture E E The group configuration eq Hyperimagi- ⇒ obtain T ; naries Utility in the unstable case expand M |= T , interpreting πE and SE accordingly eq Positive Logic eq n M M M ⇒ obtain M = hM, M /E,... ; R , f , . . . , πE ,...i. Conclusion What we have learnt Left outs Definable and algebraic closure

Imaginaries in Model Theory Definition Martin Hils Let B ⊆ U be a set of parameters and a ∈ U. Introduction a is definable over B if {a} is a B-definable set; Imaginary Galois theory a is algebraic over B if there is a finite B-definable set Algebraic closure in Meq containing a. Elimination of imaginaries The definable closure of B is given by Geometric stability Trichotomy Conjecture dcl(B) = {a ∈ U | a definable over B}. The group configuration Hyperimagi- Similarly define acl(B), the algebraic closure of B. naries Utility in the unstable case eq Positive Logic These definitions make sense in U ; eq eq eq Conclusion may write dcl or acl to stress that we work in U . What we have learnt Left outs Galois Characterisation of Algebraic Elements

Imaginaries in Model Theory

Martin Hils

Introduction

Imaginary Fact Galois theory Algebraic Let AutB (U) = {σ ∈ Aut(U) | σ(b) = b ∀ b ∈ B}. closure in Meq Elimination of 1 a ∈ dcl(B) if and only if σ(a) = a for all σ ∈ AutB (U) imaginaries Geometric 2 a ∈ acl(B) if and only if there is a finite set A0 containing stability Trichotomy a which is fixed set-wise by every σ ∈ AutB (U). Conjecture The group configuration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Existence of codes for definable sets in U eq

Imaginaries in Model Theory Fact Martin Hils For any definable set D ⊆ U n there exists c ∈ U eq (unique up Introduction to interdefinability) such that σ ∈ Aut(U) fixes D setwise if and Imaginary Galois theory only if it fixes c. Algebraic closure in Meq Elimination of imaginaries Proof. Geometric stability Suppose D is defined by ϕ(x, d). Define the equivalence Trichotomy 0 Conjecture relation E(z, z ) as The group configuration 0 Hyperimagi- ∀x(ϕ(x, z) ⇔ ϕ(x, z )). naries Utility in the unstable case Positive Logic Then c := d/E serves as a code for D. Conclusion What we have learnt Left outs The Galois Group

Imaginaries eq in Model Any σ ∈ AutB (U) ﬁxes acl (B) setwise. Theory Deﬁne the Galois group of B as Martin Hils Gal(B) := {σ acleq(B) | σ ∈ AutB (U)}. Introduction

Imaginary Galois theory Algebraic Example closure in Meq Elimination of imaginaries Let b1 6= b2 be in an infinite set without structure, Geometric stability b := (b1, b2)/E2 (think of b as {b1, b2}) and B = {b}. Trichotomy eq Conjecture Then bi ∈ acl (B) and Gal(B) = {id, σ}' Z/2, where σ The group configuration permutes b1 and b2. Hyperimagi- naries Let M |= ACF = T and K ⊆ M a subfield. Then Utility in the alg unstable case Gal(K) = Gal(K /K), where Positive Logic K alg = (field theoretic) algebraic closure of K, Conclusion alg What we Gal(K /K) = (field theoretic) Galois group of K. have learnt Left outs Galois Correspondence in T eq

Imaginaries in Model Gal(B) is a proﬁnite group: a clopen subgroup is given by Theory

Martin Hils {σ | σ(ai ) = ai ∀i}

Introduction eq Imaginary for some ﬁnite subset {a1,..., an} of acl (B). Galois theory Algebraic closure in Meq Theorem (Poizat) Elimination of imaginaries There is a 1:1 correspondence between Geometric stability Trichotomy closed subgroups of Gal(B) and Conjecture The group eq eq conﬁguration dcl -closed sets A with B ⊆ A ⊆ acl (B). Hyperimagi- naries It is given by Utility in the unstable case eq Positive Logic H 7→ {a ∈ acl (B) | h(a) = a ∀ h ∈ H}. Conclusion What we have learnt Left outs Elimination of Imaginaries

Imaginaries in Model Deﬁnition Theory

Martin Hils The theory T eliminates imaginaries if every imaginary element a ∈ U eq is interdeﬁnable with a real tuple b ∈ U n. Introduction

Imaginary Galois theory Algebraic Fact closure in Meq Elimination of imaginaries Suppose that for every ∅-definable equivalence relation E n Geometric on U there is an ∅-definable function stability Trichotomy Conjecture n m The group f : U → U (for some m ∈ N) configuration Hyperimagi- 0 0 naries such that aEa if and only if f (a) = f (a ). Utility in the unstable case Positive Logic Then T eliminates imaginaries. Conclusion The converse is (almost) true. What we have learnt Left outs Examples of theories which eliminate imaginaries

Imaginaries in Model Theory Example Martin Hils eq Introduction The theory T eliminates imaginaries. (By construction.) Imaginary The theory of an infinite set does not eliminate imaginaries. Galois theory Algebraic (The two element set {a, b} cannot be coded.) closure in Meq Elimination of Th(h , +, ×i) eliminates imaginaries. imaginaries N Geometric Algebraically closed fields eliminate imaginaries (Poizat). stability Trichotomy Many other theories of fields eliminate imaginaries. Conjecture The group configuration Hyperimagi- naries Illustration: how to code finite sets in fields? Utility in the unstable case Positive Logic Use symmetric functions: D = {a, b} is coded by the tuple 2 Conclusion (a + b, ab), as a and b are the roots of X − (a + b)X + ab. What we have learnt Left outs Utility of Elimination of Imaginaries

Imaginaries in Model Theory

Martin Hils

Introduction T has e.i. ⇒ many constructions may be done in T : Imaginary quotient objects are present in U; Galois theory Algebraic closure in codes for deﬁnable sets exist in U; Meq Elimination of imaginaries get a Galois correspondence in T eq eq Geometric (replacing dcl , acl by dcl and acl, respectively); stability Trichotomy eq Conjecture may replace T by T in the group constructions we will The group conﬁguration present in the next section. Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Main Theorem of Galois Theory

Imaginaries in Model Theory

Martin Hils

Introduction Corollary Imaginary alg Galois theory Let K be a (perfect) field and Gal(K /K) its Galois group. Algebraic closure in Then the map Meq Elimination of imaginaries H 7→ {a ∈ K alg | h(a) = a ∀ h ∈ H} Geometric stability Trichotomy Conjecture is a 1:1 correspondence between the set of closed subgroups The group alg configuration of Gal(K) and the set of intermediate fields K ⊆ L ⊆ K . Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Uncountably Categorical Theories

Imaginaries in Model Theory

Martin Hils Deﬁnition Introduction Let κ be a cardinal. A theory T is κ-categorical if, up to Imaginary Galois theory isomorphism, T has only one model of cardinality κ. Algebraic closure in Meq Elimination of imaginaries Theorem (Morley’s Categoricity Theorem) Geometric stability Trichotomy If T is κ-categorical for some uncoutable cardinal κ, then it is Conjecture The group λ-categorical for all uncountable λ. conﬁguration Hyperimagi- naries This result marks the beginning of modern model theory! Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Strongly Minimal Theories

Imaginaries in Model Theory Definition Martin Hils A definable set D is strongly minimal if for every Introduction definable subset X ⊆ D either X or D \ X is finite. Imaginary Galois theory A theory T is strongly minimal if x = x defines a Algebraic closure in Meq strongly minimal set. Elimination of imaginaries Geometric stability Trichotomy Example (strongly minimal theories) Conjecture The group configuration Hyperimagi- 1 Infinite sets without structure. naries Utility in the 2 Infinite vector spaces over some fixed field K. unstable case Positive Logic 3 Algebraically closed fields. Conclusion What we have learnt Left outs Relation to Uncountable Categoricity

Imaginaries in Model Theory

Martin Hils

Introduction Fact Imaginary Galois theory Algebraic closure in 1 Strongly minimal theories are uncountably categorical. Meq Elimination of imaginaries 2 Let T be an uncountably categorical theory. Then there is Geometric a strongly minimal set D deﬁnable in T such that T is stability Trichotomy largely controlled by D. Conjecture The group conﬁguration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Linear dependence in vector spaces

Imaginaries in Model Theory

Martin Hils V a vector space over the ﬁeld K

Introduction For A ⊆ V consider the linear span

Imaginary Galois theory ( n ) Algebraic X closure in Span(A) = ki · ai | ki ∈ K, ai ∈ A Meq Elimination of i=1 imaginaries Geometric stability X ⊆ V is linearly independent if Trichotomy Conjecture x 6∈ Span(X \{x}) for all x ∈ X The group conﬁguration X is a basis if it is maximal indep. (⇔ minimal generating) Hyperimagi- naries The dimension of V is the cardinality of a basis of V Utility in the unstable case (well-deﬁned) Positive Logic Conclusion What we have learnt Left outs acl-dependence in strongly minimal sets

Imaginaries in Model Theory

Martin Hils

Introduction Fact Imaginary Inﬁnite vector spaces are strongly minimal, with Galois theory Algebraic acl(A) = Span(A). closure in Meq Elimination of imaginaries In any strongly minimal theory, we get Geometric stability a dependence relation (and a combinatorial geometry), Trichotomy Conjecture The group using acl instead of Span; conﬁguration Hyperimagi- corresponding notions of basis and dimension. naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Geometries in strongly minimal theories

Imaginaries in Model Theory

Martin Hils 1 Inﬁnite set without structure, has a trivial geometry, i.e. pairwise independence ⇒ independence. Introduction

Imaginary Galois theory 2 a Vector spaces, are modular: Algebraic acl-closed sets A, B are independent over A ∩ B, i.e. closure in Meq Elimination of imaginaries dim(A ∪ B) = dim(A) + dim(B) − dim(A ∩ B). Geometric stability (The associated geometry is projective geometry.) Trichotomy Conjecture The group b Affine spaces, are locally modular, i.e. configuration Hyperimagi- become modular after naming some constant. naries Utility in the unstable case 3 Algebraically closed fields, are non-locally modular. Positive Logic Conclusion What we have learnt Left outs Zilber’s Trichotomy Conjecture

Imaginaries in Model Theory Guiding principle of geometric stability theory Martin Hils Geometric complexity comes from algebraic structures Introduction (e.g. infinite groups or fields) definable in the theory. Imaginary Galois theory Algebraic closure in Meq Elimination of Conjecture (Zilber) imaginaries Geometric Let T be strongly minimal. Then there are three cases: stability Trichotomy 1 Conjecture T has a trivial geometry. The group eq configuration (This implies: 6 ∃ infinite definable groups in T .) Hyperimagi- 2 naries T is locally modular non-trivial. Then a s.m. group is Utility in the eq unstable case definable in T , and its geometry is projective or affine. Positive Logic 3 If T is not locally modular, an ACF is definable in T eq. Conclusion What we have learnt Left outs Results on the Trichotomy Conjecture

Imaginaries in Model Theory

Martin Hils

Introduction Imaginary 1 True for T is totally categorical. (Zilber, late 70’s) Galois theory Algebraic 2 locally modular closure in True T for . (Hrushovski, late 80’s) Meq Elimination of 3 The conjecture is false in general. (Hrushovski 1988) imaginaries Geometric 4 True for Zariski geometries, an important special case. stability Trichotomy (Hrushovski-Zilber 1993) Conjecture The group conﬁguration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Construction of a group

Imaginaries in Model Theory Let a, b be independent elements in a strongly minimal group Martin Hils (G, ·) and c = a · b. Then

Introduction (∗) The set {a, b, c} is pairwise independent and Imaginary Galois theory dependent. Algebraic closure in Meq Elimination of imaginaries If T is non-trivial, adding some constants if necessary, Geometric stability there is a set {a, b, c} satisfying (∗). Trichotomy Conjecture The group If T is modular, any {a, b, c} satisfying (∗) comes from a conﬁguration s.m. group (G, ·) in T eq, up to interalgebraicity: Hyperimagi- naries 0 0 0 0 0 Utility in the There exist a , b ∈ G and c = a · b such that unstable case 0 0 0 Positive Logic a and a are interalgebraic, similarly b, b and c, c . Conclusion What we have learnt Left outs The Group Conﬁguration in Stable Theories

Imaginaries in Model Theory

Martin Hils

Introduction Group conﬁguration: a conﬁguration of (in-)dependences

Imaginary between tuples in U, more complicated than (∗). Galois theory Algebraic (Hrushovski) Up to interalgebraicity, any group closure in Meq eq Elimination of configuration comes from a definable group in U . imaginaries Geometric This holds in any stable theory; it is a key device in stability Geometric Stability Theory. Trichotomy Conjecture The group Source of many applications of model theory to other configuration Hyperimagi- branches of mathematics. naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Stable theories

Imaginaries in Model Theory

Martin Hils

Introduction Uncountably categorical theories are stable. Imaginary Galois theory Stable theories carry a nice notion of independence Algebraic closure in (generalising acl-independence in s.m. theories). Meq Elimination of imaginaries Stable = "no infinite set is ordered by a formula" Geometric stability The theory of any module is stable. Trichotomy Conjecture The theory of hN, +i is unstable The group configuration (x ≤ y is defined by ∃z x + z = y). Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Modularity in Stable Theories

Imaginaries in Model Theory Martin Hils Definition eq eq Introduction T stable is called modular if any acl -closed subsets A, B of Imaginary U eq are independent over their intersection A ∩ B. Galois theory Algebraic closure in Meq Elimination of This is the right notion of modularity: imaginaries Geometric Theorem stability Trichotomy eq Conjecture Let T be stable and modular . The group configuration Non-trivial dependence ⇒ ∃ infinite def. group in T eq. Hyperimagi- naries Def. groups in T eq are module-like (Hrushovski-Pillay). Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Local modularity equals modularityeq

Imaginaries in Model Theory For T strongly minimal: locally modular ⇔ modulareq. Martin Hils

Introduction Example (Aﬃne Space) Imaginary eq eq Galois theory Let L1, L2 be distinct parallel lines. Put Li = acl (Li ). Then Algebraic closure in eq eq Meq L ∩ L = ∅ and there exists a vector 0 6= v ∈ L ∩ L Elimination of 1 2 1 2 imaginaries dim(L ∪ L ) < dim(L ) + dim(L ) − dim(L ∩ L ), Geometric 1 2 1 2 1 2 stability car 3<2+2-0 Trichotomy Conjecture (⇒ non-modularity) The group conﬁguration eq eq eq eq eq eq Hyperimagi- dim(L1 ∪ L2 ) = dim(L1 ) + dim(L2 ) − dim(L1 ∩ L2 ), naries car 3 = 2 + 2 − 1 Utility in the unstable case eq Positive Logic (⇒ modularity ) Conclusion What we have learnt Left outs The notion of a hyperimaginary

Imaginaries in Model Theory Martin Hils Deﬁnition

Introduction An equivalence relation E(x, y) (where x, y are tuples of Imaginary Galois theory the same length) is said to be type-definable if Algebraic closure in Meq ^ Elimination of imaginaries xEy ⇔ ϕi (x, y) Geometric i∈N stability Trichotomy Conjecture for some sequence of L-formulas (ϕi )i∈ . The group N configuration A hyperimaginary element is an equivalence class a/E, Hyperimagi- naries for some type-definable equivalence relation E. Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs An example: monads

Imaginaries in Model Theory

Martin Hils Example

Introduction R = hR, +, ×, 0, 1,

Imaginaries in Model Theory Martin Hils Simple theories generalise stable theories;

Introduction have a good independence notion; Imaginary Galois theory simple unstable: random graph, pseudofinite fields Algebraic closure in (idea: simple = stable + some random noise). Meq Elimination of imaginaries Geometric Theorem (Ben Yaacov–Tomaˇsić–Wagner 2004) stability Trichotomy Conjecture The group configuration theorem holds in simple theories. The group configuration The corresponding group may be found in (almost) Hyperimagi- naries hyperimaginaries. Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Intrinsic Infinitesimals

Imaginaries in Model 1 Theory The example S is not an accident... Indeed

Martin Hils Theorem (2006, involves many people) Introduction ∗ Imaginary Let G be a definable compact group in R < R Galois theory ∗ Algebraic (or more generally in an o-minimal expansion of R ). closure in Meq Elimination of imaginaries 1 There is a type-definable subgroup µ ≤ G Geometric stability ⇒ cosets g · µ are hyperimaginaries. Trichotomy Conjecture 2 ∗ The group The group (G/µ)(R ) is isomorphic to a group over the configuration standard real numbers R and shares many properties with Hyperimagi- naries G (e.g. has the same dimension). Utility in the unstable case 3 Positive Logic µ gives rise to an intrinsic notion of monad. Conclusion What we have learnt Left outs Losing Compactness on Hyperimaginary Sorts

Imaginaries in Model Theory One would like to add sorts for hyperimaginaries to L. Martin Hils Example (back to the unit circle) Introduction Imaginary S1 in R, with xEy ⇔ V dist(x, y) < 1 ; Galois theory n∈N n Algebraic 1 closure in eq S /E is infinite, but bounded, since it does not grow in M ∗ Elimination of elementary extensions R R; imaginaries < Geometric ⇒ Compactness is violated if a sort for S1/E is added in stability Trichotomy first order logic: Conjecture The group configuration 1 Hyperimagi- {x/E 6= a/E | a ∈ S (R)} naries Utility in the unstable case is finitely satisfiabe but unsatisfiable. Positive Logic Conclusion What we have learnt Left outs Adding Hyperimaginary Sorts in Positive Logic

Imaginaries in Model Theory In fact, negation is the only obstacle:

Martin Hils Theorem (Ben Yaacov) Introduction

Imaginary One may add sorts for hyperimaginaries in positive logic without Galois theory Algebraic losing compactness. closure in Meq Elimination of imaginaries eq Geometric This is similar to Shelah’s M -construction. stability Trichotomy On a hyperimaginary sort D/E, add predicates for any Conjecture The group conﬁguration subset X ⊆ D/E such that Hyperimagi- naries π−1(X ) = {d ∈ D | d/E ∈ X } Utility in the unstable case Positive Logic Conclusion is type-deﬁnable without parameters. What we have learnt Left outs Where to look

Imaginaries in Model Theory

Martin Hils

Introduction Imaginaries are needed in order to

Imaginary Galois theory 1 understand independence, modularity etc.; Algebraic closure in Meq 2 get a decent Galois correspondence; Elimination of imaginaries Geometric 3 find algebraic structures like infinite groups or fields, stability explaining a complicated geometric behaviour. Trichotomy Conjecture The group configuration ⇒ Need to classify imaginaries to fully understand T . Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Beyond (ordinary) imaginaries

Imaginaries in Model Theory

Martin Hils

Introduction

Imaginary In more general contexts one might have to Galois theory Algebraic closure in 1 go even beyond imaginaries; Meq Elimination of imaginaries 2 consider hyperimaginaries or more complicated objects; Geometric stability 3 adapt the logical framework (⇒ positive logic). Trichotomy Conjecture The group conﬁguration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs Important left outs

Imaginaries in Model Theory

Martin Hils

Introduction Imaginary 1 The use of imaginaries to analyse types (or groups) by Galois theory Algebraic breaking them down into irreducible ones (e.g. rank 1). closure in Meq Elimination of 2 Groupoid imaginaries. imaginaries Geometric 3 The recent classification of imaginaries in algebraically stability Trichotomy closed valued fields (Haskell–Hrushovski–Macpherson). Conjecture The group configuration Hyperimagi- naries Utility in the unstable case Positive Logic Conclusion What we have learnt Left outs