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Author Description

Ehud “Udi” Hrushovski is an Israeli mathematician renowned for his work in model theory and its applications to algebraic geometry, number theory, and combinatorics. He has held positions at institutions such as the Hebrew University of Jerusalem and the University of Oxford. Among his notable contributions is the so-called “Hrushovski construction,” a technique in model theory that has led to new insights in various branches of mathematics. He has received several major awards and honors for his research, reflecting his status as one of the leading figures in the field.

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A new strongly minimal set

1993-07-01
Ehud Hrushovski

Model theory of difference fields

1999-04-08
Zoé Chatzidakis , Ehud Hrushovski
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.

The Mordell-Lang conjecture for function fields

1996-01-01
Ehud Hrushovski
We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in … We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Groups, measures, and the NIP

2007-02-02
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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Stable group theory and approximate subgroups

2011-06-15
Ehud Hrushovski
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math … We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
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On NIP and invariant measures

2011-05-13
Ehud Hrushovski , Anand Pillay
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
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Extending partial isomorphisms of graphs

1992-12-01
Ehud Hrushovski

Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10
Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson
This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and … This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.
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The Elementary Theory of the Frobenius Automorphisms

2004-06-25
Ehud Hrushovski
A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of … A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.

Finite Structures with Few Types

1993-01-01
Ehud Hrushovski

The Manin–Mumford conjecture and the model theory of difference fields

2001-09-01
Ehud Hrushovski

Groups definable in local fields and pseudo-finite fields

1994-02-01
Ehud Hrushovski , Anand Pillay

Integration in valued fields

2007-12-31
Ehud Hrushovski , David Kazhdan
We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue … We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.
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Strongly minimal expansions of algebraically closed fields

1992-10-01
Ehud Hrushovski

Locally modular regular types

1987-01-01
Ehud Hrushovski

Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01
Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain … It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.

Non-archimedean tame topology and stably dominated types

2010-01-01
Ehud Hrushovski , François Loeser
Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the … Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it. In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$. When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values. The space $\hat {V}$ is obtained by defining a topology on the pro-definable set of stably dominated types on $V$. The key result is the construction of a pro-definable strong retraction of $\hat {V}$ to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.
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Unidimensional theories are superstable

1990-11-01
Ehud Hrushovski

Zariski geometries

1996-01-01
Ehud Hrushovski , Boris Zilber
A characterization is obtained of the Zariski topology over an algebraically closed field. A characterization is obtained of the Zariski topology over an algebraically closed field.
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Generically stable and smooth measures in NIP theories

2012-12-13
Ehud Hrushovski , Anand Pillay , Pierre Simon
We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by … We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.
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Polynomial Invariants for Affine Programs

2018-06-27
Ehud Hrushovski , Joël Ouaknine , Amaury Pouly +1 more
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed … We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.
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MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

2002-08-02
Zoé Chatzidakis , Ehud Hrushovski , Ya’acov Peterzil
We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the … We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

Almost orthogonal regular types

1989-12-01
Ehud Hrushovski

Non-Archimedean Tame Topology and Stably Dominated Types

2016-03-18
Ehud Hrushovski , François Loeser
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model … Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p -adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.
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Totally categorical structures

1989-01-01
Ehud Hrushovski
A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is … A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.
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On central extensions and definably compact groups in o-minimal structures

2010-11-17
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
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Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

2018-07-18
Ehud Hrushovski , Benjamin Martin , Silvain Rideau
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . … We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p ) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
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Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

2004-02-27
Ehud Hrushovski , Itamar Pitowsky
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Unimodular Minimal Structures

1992-12-01
Ehud Hrushovski

Groupoids, imaginaries and internal covers

2012-01-01
Ehud Hrushovski
Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of … Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.
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Difference fields and descent in algebraic dynamics. I

2008-10-01
Zoé Chatzidakis , Ehud Hrushovski
Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In … Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line. The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.
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Kueker's conjecture for stable theories

1989-03-01
Ehud Hrushovski
Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results … Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.

Groupoids, imaginaries and internal covers

2012-04-25
Ehud Hrushovski
Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of … Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.

Zariski geometries

1993-01-01
Ehud Hrushovski , Boris Zilber
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets. We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
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On model complete differential fields

2003-07-08
Ehud Hrushovski , Masanori Itai
We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of … We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>. We show that this sub-family is usually definable (in particular if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

Monodromy and the Lefschetz fixed point formula

2015-01-01
Ehud Hrushovski , François Loeser
We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy … We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.
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Motivic Poisson Summation

2009-01-01
Ehud Hrushovski , David Kazhdan
We develop a "motivic integration" version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums.We also study division algebras over the … We develop a "motivic integration" version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums.We also study division algebras over the function field, and obtain relations among the motivic Fourier transforms of a test function at different completions.We use these to prove, in a special case, a motivic version of a theorem of [DKV].
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Projective Planes in Algebraically Closed Fields

1991-01-01
David M. Evans , Ehud Hrushovski
We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. … We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particular configuration of four lines and six points in the geometry indicates the presence of a connected one-dimensional algebraic group.

Valued fields, metastable groups

2019-07-23
Ehud Hrushovski , Silvain Rideau
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Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties

2000-06-01
Ehud Hrushovski , Anand Pillay
Let A be a semi-abelian variety, and X a subvariety of A , both defined over a number field. Assume that X does not contain X 1 + X 2 … Let A be a semi-abelian variety, and X a subvariety of A , both defined over a number field. Assume that X does not contain X 1 + X 2 for any positive-dimensional subvarieties X 1 , X 2 of A . Let Γ be a subgroup of A ( C ) of finite rational rank. We give doubly exponential bounds for the size of ( X ∩ Γ)\ X ( Ǭ ). Among the ingredients is a uniform bound, doubly exponential in the data, on finite sets which are quantifier-free definable in differentially closed fields. We also give uniform bounds on X ∩ Γ in the case where X contains no translate of any semi-abelian subvariety of A and Γ is a subgroup of A ( C ) of finite rational rank which has trivial intersection with A ( Ǭ ). (Here A is assumed to be defined over a number field, but X need not be.)

The Value Ring of Geometric Motivic Integration, and the Iwahori Hecke Algebra of SL 2

2008-02-18
Ehud Hrushovski , David Kazhdan , Nir Avni
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The Uncountable Spectra of Countable Theories

2000-07-01
Bradd Hart , Ehud Hrushovski , M. Laskowski
Let T be a complete, first-order theory in a finite or countable language having infinite models.Let I(T, κ) be the number of isomorphism types of models of T of cardinality … Let T be a complete, first-order theory in a finite or countable language having infinite models.Let I(T, κ) be the number of isomorphism types of models of T of cardinality κ.We denote by µ (respectively μ) the number of cardinals (respectively infinite cardinals) less than or equal to κ.Theorem.I(T, κ), as a function of κ > ℵ 0 , is the minimum of 2 κ and one of the following functions:4. the constant function 2 ;5. d+1 (µ) for some infinite, countable ordinal d;6. d i=1 Γ(i) where d is an integer greater than 0 (the depth of T ) andwhere σ(i) is either 1, ℵ 0 or 1 , and α(i) is 0 or 2 ; the first possibility for Γ(i) can occur only when di > 0.The cases (2), (3) of functions taking a finite value were dealt with by Morley and Lachlan.Shelah showed (1) holds unless a certain structure theory (superstability and extensions) is valid.He also characterized (4) and ( 5) and
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On the automorphism groups of finite covers

1993-07-01
David M. Evans , Ehud Hrushovski

Berkovich spaces embed in Euclidean spaces

2015-04-27
Ehud Hrushovski , François Loeser , Bjorn Poonen
Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an … Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an of any d-dimensional quasi-projective scheme V over K embeds in R^{2d+1}. If, moreover, the value group of K is dense in R_{>0} and V is a curve, then we describe the homeomorphism type of V^an by using the theory of local dendrites.
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A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007-03-01
Ehud Hrushovski , Ya’acov Peterzil
Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions … Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.
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A dichotomy theorem for regular types

1989-12-01
Ehud Hrushovski , Saharon Shelah

Counting and dimensions

2008-05-22
Ehud Hrushovski , Frank Olaf Wagner
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Zeta functions from definable equivalence relations

2006-12-31
Ehud Hrushovski , Benjamin Martin
We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also … We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.

Finitely based theories

1989-03-01
Ehud Hrushovski
Abstract A stable theory is finitely based if every set of indiscernibles is based on a finite subset. This is a common generalization of superstability and 1-basedness. We show that … Abstract A stable theory is finitely based if every set of indiscernibles is based on a finite subset. This is a common generalization of superstability and 1-basedness. We show that if such theories have more than one model they must have infinitely many, and prove some other conjectures.

On Pseudo-Finite Dimensions

2013-01-01
Ehud Hrushovski
We attempt to formulate issues around modularity and Zilber's trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9]. We attempt to formulate issues around modularity and Zilber's trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].
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Compactifications of pseudofinite and pseudoamenable groups

2025-01-14
Gabriel Conant , Ehud Hrushovski , Anand Pillay
We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) … We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.
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Globally valued fields: foundations

2024-09-06
Itaï Ben Yaacov , Pablo Destic , Ehud Hrushovski +1 more
We present foundations of globally valued fields, i.e., of a class of fields with an extra structure, capturing some aspects of the geometry of global fields, based on the product … We present foundations of globally valued fields, i.e., of a class of fields with an extra structure, capturing some aspects of the geometry of global fields, based on the product formula. We provide a dictionary between various data defining such extra structure: syntactic (models of some unbounded continuous logic theory), Arakelov theoretic, and measure theoretic. In particular we obtain a representation theorem relating globally valued fields and adelic curves defined by Chen and Moriwaki.
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Approximate equivalence relations

2024-07-19
Ehud Hrushovski
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Revisiting virtual difference ideals

2024-07-19
Zoé Chatzidakis , Ehud Hrushovski
In difference algebra, basic definable sets correspond to prime ideals that are invariant under a structural endomorphism.The main idea of [5] was that periodic prime ideals enjoy better geometric properties … In difference algebra, basic definable sets correspond to prime ideals that are invariant under a structural endomorphism.The main idea of [5] was that periodic prime ideals enjoy better geometric properties than invariant ideals; and to understand a definable set, it is helpful to enlarge it by relaxing invariance to periodicity, obtaining better geometric properties at the limit.The limit in question was an intriguing but somewhat ephemeral setting called virtual ideals.However a serious technical error was discovered by Tom Scanlon's UCB seminar.In this text, we correct the problem via two different routes.We replace the faulty lemma by a weaker one, that still allows recovering all results of [5] for all virtual ideals.In addition, we introduce a family of difference equations ("cumulative" equations) that we expect to be useful more generally.Results in [4] imply that cumulative equations suffice to coordinatize all difference equation.For cumulative equations, we show that virtual ideals reduce to globally periodic ideals, thus providing a proof of Zilber's trichotomy for difference equations using periodic ideals alone.
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Approximate equivalence relations

2024-06-27
Ehud Hrushovski
We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. … We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim-Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov-Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an 'approximately unique' probability logic 1-type, we obtain a structure theorem generalizing the `Lie model' theorem for approximate subgroups. The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. For NIP theories, pursuing a question of Pillay's, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets.
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Tropical functions on a skeleton

2024-06-24
Antoine Ducros , Ehud Hrushovski , François Loeser +1 more
We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework … We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a model-theoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence.
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Model Equivalences

2024-06-21
Michael Benedikt , Ehud Hrushovski
We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, … We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of definable MERs, along with the first steps of a classification theory for them. We characterize the special classes of definable MERs associated with preservation of formulas, either in classical first order logic or in continuous logic, and uncover an intrinsic role for the latter. We bring out a nontrivial relationship with interpretations (imaginary sorts), leading to a wider hierarchy of classes related to the preservation of reducts. We give results about the relationship between these classes, both for general theories and for theories satisfying additional model-theoretic properties, such as stability.
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Lang-Weil Type Estimates in Finite Difference Fields

2024-06-02
Martin Hils , Ehud Hrushovski , J. Ye +1 more
We prove a uniform estimate of the number of points for difference algebraic varieties in finite difference fields in the spirit of Lang-Weil. More precisely, we give uniform lower and … We prove a uniform estimate of the number of points for difference algebraic varieties in finite difference fields in the spirit of Lang-Weil. More precisely, we give uniform lower and upper bounds for the number of rational points of a difference variety in terms of its transformal dimension. As a main technical ingredient, we prove an equidimensionality result for Frobenius reductions of difference varieties.
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On metric approximate subgroups

2024-05-31
Ehud Hrushovski , Arturo Rodríguez Fanlo
Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around … Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of [Formula: see text] finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].
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The Theory of the Entire Algebraic Functions

2024-01-12
Taylor Dupuy , Ehud Hrushovski
Abstract Let $A$ be the integral closure of the ring of polynomials ${{\mathbb {C}}}[t]$, within the field of algebraic functions in one variable. We show that $A$ interprets the ring … Abstract Let $A$ be the integral closure of the ring of polynomials ${{\mathbb {C}}}[t]$, within the field of algebraic functions in one variable. We show that $A$ interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory in [12] and [4].
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Embedded Finite Models beyond Restricted Quantifier Collapse

2023-06-26
Michael Benedikt , Ehud Hrushovski
We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded … We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded finite model theory. We extend the analysis of "collapse results": the ability to eliminate first-order quantifiers over the infinite domain in favor of quantification over the finite structure. We investigate several weakenings of collapse, one allowing higher-order quantification over the finite structure, another allowing expansion of the theory. We also provide results comparing collapse for unary signatures with general signatures, and new analyses of collapse for natural decidable theories.
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Embedded Finite Models beyond Restricted Quantifier Collapse

2023-01-01
Michael Benedikt , Ehud Hrushovski
We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded … We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded finite model theory. We extend the analysis of "collapse results": the ability to eliminate first-order quantifiers over the infinite domain in favor of quantification over the finite structure. We investigate several weakenings of collapse, one allowing higher-order quantification over the finite structure, another allowing expansion of the theory. We also provide results comparing collapse for unary signatures with general signatures, and new analyses of collapse for natural decidable theories.
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The Theory of the Entire Algebraic Functions

2023-01-01
Taylor Dupuy , Ehud Hrushovski
Let A be the integral closure of the ring of polynomials CC[t], within the field of algebraic functions in one variable. We show that A interprets the ring of integers. … Let A be the integral closure of the ring of polynomials CC[t], within the field of algebraic functions in one variable. We show that A interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory (see Prestel-Schmid and van den Dries-A. Macintyre).

Compactifications of pseudofinite and pseudo-amenable groups

2023-01-01
Gabriel Conant , Ehud Hrushovski , Anand Pillay
We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a … We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.
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Imaginaries, products and the adele ring

2023-01-01
Jamshid Derakhshan , Ehud Hrushovski
We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in … We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.

Invariant measures in simple and in small theories

2022-08-08
Artem Chernikov , Ehud Hrushovski , Alex Kruckman +4 more
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism … We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups.
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Specialization of Difference Equations and High Frobenius Powers

2022-01-01
Yuval Dor , Ehud Hrushovski
We study valued fields equipped with an automorphism $\sigma$ which is locally infinitely contracting in the sense that $\alpha\ll\sigma\alpha$ for all $0<\alpha\in\Gamma$. We show that various notions of valuation theory, … We study valued fields equipped with an automorphism $\sigma$ which is locally infinitely contracting in the sense that $\alpha\ll\sigma\alpha$ for all $0<\alpha\in\Gamma$. We show that various notions of valuation theory, such as Henselian and strictly Henselian hulls, admit meaningful transformal analogues. We prove canonical amalgamation results, and exhibit the way that transformal wild ramification is controlled by torsors over generalized vector groups. Model theoretically, we determine the model companion: it is decidable, admits a simple axiomatization, and enjoys elimination of quantifiers up to algebraically bounded quantifiers. The model companion is shown to agree with the limit theory of the Frobenius action on an algebraically closed and nontrivially valued field. This opens the way to a motivic intersection theory for difference varieties that was previously available only in characteristic zero. As a first consequence, the class of algebraically closed valued fields equipped with a distinguished Frobenius $x\mapsto x^{q}$ is decidable, uniformly in $q$.
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Globally valued function fields: existential closure

2022-01-01
Itaï Ben Yaacov , Ehud Hrushovski
These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em … These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof that {\em the canonical GVF structure on $k(t)^{alg}$ is existentially closed}. This can be read as saying that a variety {\em with a distinguished curve class} is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields.
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Amenability, connected components, and definable actions

2021-12-08
Ehud Hrushovski , Krzysztof Krupiński , Anand Pillay
Abstract We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an … Abstract We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>000</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group topologies on a given $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>000</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for any model M . Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G -invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> in a suitable language (where $${{\mathbb {F}}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is the free group in 2-generators) for which the $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group $$H:=\langle X \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>X</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).
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Piecewise Interpretable Hilbert Spaces

2021-10-11
Alexis Chevalier , Ehud Hrushovski
We define and study piecewise interpretable Hilbert spaces in continuous logic. These are Hilbert spaces which arise as direct limits of imaginary sorts of a model $M$ of a theory … We define and study piecewise interpretable Hilbert spaces in continuous logic. These are Hilbert spaces which arise as direct limits of imaginary sorts of a model $M$ of a theory $T$. We introduce natural examples of piecewise interpretable Hilbert spaces in a wide variety of contexts. We show that piecewise interpretable Hilbert spaces can be seen as encoding interesting model theoretic information about $M$ or $T$, such as properties of definable measures or Galois-theoretic information. We also show that piecewise interpretable Hilbert spaces encode unitary group representations in various ways. We carry out a systematic structural analysis of piecewise interpretable Hilbert spaces with scattered subsets. As an application of this work, we recover the classification of the unitary representations of oligomorphic groups first discovered by Tsankov (2012). Our main tool is local stability theory in continuous logic.

Ax’s theorem with an additive character

2021-08-31
Ehud Hrushovski
Motivated by Emmanuel Kowalski’s exponential sums over definable sets in finite fields, we generalize Ax’s theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role … Motivated by Emmanuel Kowalski’s exponential sums over definable sets in finite fields, we generalize Ax’s theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played byWeil’s Riemann hypothesis for curves over finite fields is taken by the ‘Weil bound’ on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis–Van den Dries–Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions \mathbb{F}^n_p \to \mathbb{C} obtained from the additive characters and the characteristic functions of subvarieties by pre- or postcomposing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime p, by a polynomial expression in \Psi_p(\xi) at certain algebraic functions \xi of the variables, where \Psi ( n mod p ) = exp( 2\pi i n /p ) is the standard additive character.
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Definable Equivariant Retractions in Non-Archimedean Geometry

2021-01-01
Martin Hils , Ehud Hrushovski , Pierre Simon
For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as … For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $\widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $\widehat{S}$ onto a definable group which is internal to the value group. We show that, in case $S$ is defined over a complete valued field $K$ with value group a subgroup of $\mathbb{R}$, this map descends to an $S(K)$-equivariant strong deformation retraction of the Berkovich analytification $S^{\mathrm{an}}$ of $S$ onto a piecewise linear group, namely onto the skeleton of $S^{\mathrm{an}}$. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of $S$. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group $G$ is a directed union of $\infty$-definable subgroups which all stabilize a generically stable Keisler measure on $G$.

Piecewise Interpretable Hilbert Spaces

2021-01-01
Alexis Chevalier , Ehud Hrushovski
We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a … We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free} components, where short-range interactions are controlled by algebraic closure and long-range interactions vanish. Examples include $L^2$-spaces relative to Macpherson-Steinhorn definable measures; $L^2$ spaces relative to the Haar measure of the absolute Galois groups; irreducible unitary representations of $p$-adic Lie groups; and unitary representations of the automorphism group of an $\omega$-categorical theory. In the last case, our main result specialises to a theorem of Tsankov. New methods are required, making essential use of local stability theory in continuous logic.
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Invariant measures in simple and in small theories

2021-01-01
Artem Chernikov , Ehud Hrushovski , Alex Kruckman +4 more
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant … We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.
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Beyond the Lascar Group

2020-11-24
Ehud Hrushovski
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, … We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({\mathbb{R}})$ or $SL_n({\mathbb{Q}}_p)$.

On first order amenability

2020-01-01
Ehud Hrushovski , Krzysztof Krupiński , Anand Pillay
We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an … We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $ω$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $ω$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.

Beyond the Lascar Group

2020-01-01
Ehud Hrushovski
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, … We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({\mathbb{R}})$ or $SL_n({\mathbb{Q}}_p)$.

Definability patterns and their symmetries

2019-11-04
Ehud Hrushovski
We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary … We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupinski, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of \cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $\beth_5$ to $\beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the Kechris-Pestov-Todor\v cevic correspondence. We define the Ramsey property for a first order theory, and show - as a simple application of the construction applied to an auxiliary theory - that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraisse classes, first by Kechris-Pestov-Todor\v cevic for expansions by orderings, then by Melleray, Nguyen Van The, Tsankov and Zucker for more general expansions.

Ax's theorem with an additive character

2019-11-04
Ehud Hrushovski
Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role … Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $\Ff_p^n \to \Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where $\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character.

Valued fields, metastable groups

2019-07-23
Ehud Hrushovski , Silvain Rideau
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Amenability, connected components, and definable actions

2019-01-09
Ehud Hrushovski , Krzysztof Krupiński , Anand Pillay
We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration … We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain ``weak Bohr compactification'' introduced in [24]. In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a ``definable-topological'' context, confirming the main conjectures from [24]. We also introduce $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. Thirdly, we give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language (where $\mathbb{F}_2$ is the free group in 2-generators) for which the $\bigvee$-definable group $H:=\langle X \rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) ``model'' exists for each approximate subgroup does not work in general (they proved in [29] that it works for definably amenable approximate subgroups).

Non-locally modular regular types in classifiable theories

2019-01-01
Élisabeth Bouscaren , Bradd Hart , Ehud Hrushovski +1 more
We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for … We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $p\in S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.
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Definability patterns and their symmetries

2019-01-01
Ehud Hrushovski
We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary … We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupi\'nski, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of \cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $\beth_5$ to $\beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the Kechris-Pestov-Todor\v cevi\'c correspondence. We define the Ramsey property for a first order theory, and show - as a simple application of the construction applied to an auxiliary theory - that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraiss\'e classes, first by Kechris-Pestov-Todor\v cevi\'c for expansions by orderings, then by Melleray, Nguyen Van Th\'e, Tsankov and Zucker for more general expansions.
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Amenability, connected components, and definable actions

2019-01-01
Ehud Hrushovski , Krzysztof Krupiński , Anand Pillay
We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results … We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjectures from [24]. We introduce $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. We study the relationship between definability of an action of a definable group on a compact space, weakly almost periodic actions, and stability. We conclude that for any group $G$ definable in a sufficiently saturated structure, every definable action of $G$ on a compact space supports a $G$-invariant probability measure. This gives negative solutions to some questions and conjectures from [22] and [24]. We give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language for which the $\bigvee$-definable group $H:=\langle X \rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact "model" exists for each approximate subgroup does not work in general.
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Ax's theorem with an additive character

2019-01-01
Ehud Hrushovski
Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role … Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $\Ff_p^n \to \Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where $\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character.
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Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

2018-07-18
Ehud Hrushovski , Benjamin Martin , Silvain Rideau
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . … We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p ) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
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Polynomial Invariants for Affine Programs

2018-06-27
Ehud Hrushovski , Joël Ouaknine , Amaury Pouly +1 more
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed … We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.
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Polynomial Invariants for Affine Programs

2018-02-06
Ehud Hrushovski , Joël Ouaknine , Amaury Pouly +1 more
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed … We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.

Polynomial Invariants for Affine Programs

2018-01-01
Ehud Hrushovski , Joël Ouaknine , Amaury Pouly +1 more
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed … We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.
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A closer look at the stable completion

2017-10-19
Ehud Hrushovski , François Loeser
This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript <italic>n</italic> in terms of spaces of semi-lattices, with particular … This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript <italic>n</italic> in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript <italic>n</italic> into the inverse limit of a system of spaces of semi-lattices <italic>L</italic>(<italic>H</italic>subscript <italic>d</italic>) endowed with the linear topology, where <italic>H</italic>subscript <italic>d</italic> are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism <italic>L</italic>(<italic>H</italic>subscript <italic>d</italic>). The chapter also considers the condition that if a definable set in <italic>L</italic>(<italic>H</italic>subscript <italic>d</italic>) is an intersection of relatively compact sets, then it is itself relatively compact.

Γ‎-internal spaces

2017-10-19
Ehud Hrushovski , François Loeser
This chapter describes the topological structure of Γ‎-internal spaces. Let <italic>V</italic> be an algebraic variety over a valued field. An iso-definable subset <italic>X</italic> of unit vector <italic>V</italic> is said to … This chapter describes the topological structure of Γ‎-internal spaces. Let <italic>V</italic> be an algebraic variety over a valued field. An iso-definable subset <italic>X</italic> of unit vector <italic>V</italic> is said to be Γ‎-internal if it is in pro-definable bijection with a definable set which is Γ‎-internal. A number of delicate issues arise here. A pro-definable subset <italic>X</italic> of unit vector V is Γ‎-parameterized if there exists a definable subset <italic>Y</italic> of Γ‎ⁿ, for some <italic>n</italic>, and a pro-definable map <italic>g</italic> : <italic>Y</italic> → unit vector V with image <italic>X</italic>. The chapter presents an example showing that there exists Γ‎-parameterized subsets of unit vector V which are not iso-definable, whence not Γ‎-internal. It also presents the main results about the topological structure of Γ‎-internal spaces.

Definable compactness

2017-10-19
Ehud Hrushovski , François Loeser
This chapter describes the notion of definable compactness for subsets of unit vector V. One of the main results is Theorem 4.2.20, which establishes the equivalence between being definably compact … This chapter describes the notion of definable compactness for subsets of unit vector V. One of the main results is Theorem 4.2.20, which establishes the equivalence between being definably compact and being closed and bounded. The chapter gives a general definition of definable compactness that may be useful when the definable topology has enough definable types. The o-minimal formulation regarding limits of curves is replaced by limits of definable types. The chapter relates definable compactness to being closed and bounded and shows that the expected properties hold. In particular, the image of a definably compact set under a continuous definable map is definably compact.

Continuity of homotopies

2017-10-19
Ehud Hrushovski , François Loeser
This chapter includes some additional material on homotopies. In particular, for a smooth variety <italic>V</italic>, there exists an “inflation” homotopy, taking a simple point to the generic type of a … This chapter includes some additional material on homotopies. In particular, for a smooth variety <italic>V</italic>, there exists an “inflation” homotopy, taking a simple point to the generic type of a small neighborhood of that point. This homotopy has an image that is properly a subset of unit vector V, and cannot be understood directly in terms of definable subsets of <italic>V</italic>. The image of this homotopy retraction has the merit of being contained in unit vector U for any dense Zariski open subset <italic>U</italic> of <italic>V</italic>. The chapter also proves the continuity of functions and homotopies using continuity criteria and constructs inflation homotopies before proving GAGA type results for connectedness. Additional results regarding the Zariski topology are given.

Curves

2017-10-19
Ehud Hrushovski , François Loeser
This chapter proves the iso-definability of unit vector C when <italic>C</italic> is a curve using Riemann-Roch. Recall that a pro-definable set is called iso-definable if it is isomorphic, as a … This chapter proves the iso-definability of unit vector C when <italic>C</italic> is a curve using Riemann-Roch. Recall that a pro-definable set is called iso-definable if it is isomorphic, as a pro-definable set, to a definable set. If <italic>C</italic> is an algebraic curve defined over a valued field <italic>F</italic>, then unit vector C is an iso-definable set. The topology on unit vector C is definably generated, that is, generated by a definable family of (iso)-definable subsets. In other words, there is a definable family giving a pre-basis of the topology. The chapter explains how definable types on <italic>C</italic> correspond to germs of paths on unit vector C. It also constructs the retraction on skeleta for curves. A key result is the finiteness of forward-branching points.

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

2017-10-19
Ehud Hrushovski , François Loeser
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model … Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

Introduction

2017-10-19
Ehud Hrushovski , François Loeser
This book deals with non-archimedean tame topology and stably dominated types. It considers o-minimality as an analogy and reduces questions over valued fields to the o-minimal setting. A fundamental tool, … This book deals with non-archimedean tame topology and stably dominated types. It considers o-minimality as an analogy and reduces questions over valued fields to the o-minimal setting. A fundamental tool, imported from stability theory, is the notion of a definable type, which plays a number of roles, starting from the definition of a point of the fundamental spaces. One of the roles of definable types is to be a substitute for the classical notion of a sequence, especially in situations where one is willing to refine to a subsequence. To each algebraic variety <italic>V</italic> over a valued field <italic>K</italic>, the book associates in a canonical way a projective limit unit vector V of spaces, which is the stable completion of <italic>V</italic>. In case the value group is ℝ, the results presented in this book relate to similar tameness theorems for Berkovich spaces.

Specializations and ACV2F

2017-10-19
Ehud Hrushovski , François Loeser
This chapter introduces the theory ACV²F of iterated places and describes some algebraic criteria for v- and g-continuity. It considers the theory ACV²F of triples (<italic>K</italic>₂,<italic>K</italic>₁,<italic>K</italic>₀) of fields with surjective, … This chapter introduces the theory ACV²F of iterated places and describes some algebraic criteria for v- and g-continuity. It considers the theory ACV²F of triples (<italic>K</italic>₂,<italic>K</italic>₁,<italic>K</italic>₀) of fields with surjective, non-injective places <italic>r</italic>ᵢⱼ : <italic>K</italic>ᵢ → <italic>K</italic>ⱼ for <italic>i</italic> &gt; <italic>j</italic>, <italic>r</italic>₂₀ = <italic>r</italic>₁₀ ° <italic>r</italic>₂₁, such that <italic>K</italic>₂ is algebraically closed. The chapter shows that the family of g-open sets is definable in definable families. It also presents some applications of the continuity criteria and concludes by proving that for each definable set of definable functions <italic>V</italic> → <italic>W</italic>superscript Number Sign the subset of those that are g-continuous is definable.

Preliminaries

2017-10-19
Ehud Hrushovski , François Loeser
This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these … This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable <italic>V</italic> as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

An equivalence of categories

2017-10-19
Ehud Hrushovski , François Loeser
This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable homotopy … This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable homotopy category of definable spaces over the o-minimal Γ‎. The chapter introduces three categories that can be viewed as ind-pro definable and admit natural functors to the category TOP of topological spaces with continuous maps. The discussion is often limited to the subcategory consisting of A-definable objects and morphisms. The morphisms are factored out by (strong) homotopy equivalence. The chapter presents the proof of the equivalence of categories before concluding with remarks on homotopies over imaginary base sets.

The smooth case

2017-10-19
Ehud Hrushovski , François Loeser
This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an <italic>F</italic>-definable homotopy … This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an <italic>F</italic>-definable homotopy <italic>h</italic> : <italic>I</italic> × unit vector X → unit vector X and the properties for <italic>h</italic>. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set <italic>V</italic>₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into <italic>V</italic>₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.
Coauthor Papers Together
Anand Pillay 27
François Loeser 21
Zoé Chatzidakis 16
Dugald Macpherson 8
David Kazhdan 7
Ya’acov Peterzil 7
Deirdre Haskell 7
Krzysztof Krupiński 6
Silvain Rideau 5
M. Laskowski 5
Pierre Simon 5
Bradd Hart 5
Élisabeth Bouscaren 3
Boris Zilber 3
Özlem Beyarslan 3
Amaury Pouly 3
James Worrell 3
Daniel Palacín 3
Michael Benedikt 3
Benjamin Martin 3
Joël Ouaknine 2
Taylor Dupuy 2
Itaï Ben Yaacov 2
Martin Hils 2
Slavko Moconja 2
Thomas Scanlon 2
Frank Olaf Wagner 2
Artem Chernikov 2
Alexis Chevalier 2
Itamar Pitowsky 2
David M. Evans 2
Saharon Shelah 2
Gabriel Conant 2
Alex Kruckman 2
Nicholas Ramsey 2
Bernhard Herwig 1
Masanori Itai 1
Yuval Dor 1
Assaf Hasson 1
Arturo Rodríguez Fanlo 1
Peter H. Kropholler 1
Aner Shalev 1
Antoine Ducros 1
Genady Ya. Grabarnik 1
Bjorn Poonen 1
Alexander Lubotzky 1
Bruno Poizat 1
Françoise Point 1
Joël Ouaknine 1
George M. Bergman 1

Model theory of difference fields

1999-04-08
Zoé Chatzidakis , Ehud Hrushovski
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.

Geometric Stability Theory

1996-09-12
Anand Pillay
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

Definable sets in algebraically closed valued fields: elimination of imaginaries

2006-01-01
Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain … It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.

Weakly Normal Groups

1987-01-01
U. Hrushovski , Anthony L. Pillay

Groups, measures, and the NIP

2007-02-02
Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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Finite Structures with Few Types

1993-01-01
Ehud Hrushovski

Unidimensional theories are superstable

1990-11-01
Ehud Hrushovski

Groups definable in local fields and pseudo-finite fields

1994-02-01
Ehud Hrushovski , Anand Pillay

Kueker's conjecture for stable theories

1989-03-01
Ehud Hrushovski
Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results … Abstract Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.

The Elementary Theory of Finite Fields

1968-09-01
James Ax

On NIP and invariant measures

2011-05-13
Ehud Hrushovski , Anand Pillay
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.
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Rigid Subanalytic Sets

1993-02-01
Leonard Lipshitz

Measures and forking

1987-01-01
H. Jerome Keisler

A survey of basic stability theory, with particular emphasis on orthogonality and regular types

1984-09-01
Michael Makkai

Integration in valued fields

2007-12-31
Ehud Hrushovski , David Kazhdan
We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue … We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.
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On strongly minimal sets

1971-03-01
John T. Baldwin , A. H. Lachlan
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. … The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

Locally modular regular types

1987-01-01
Ehud Hrushovski

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01
Ya’acov Peterzil , Charles Steinhorn
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Continuous first order logic and local stability

2010-06-10
Itaï Ben Yaacov , Alexander Usvyatsov
We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open … We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
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Strongly minimal countably categorical theories. II

1985-01-01
Boris Zilber

The Manin–Mumford conjecture and the model theory of difference fields

2001-09-01
Ehud Hrushovski

ℵ0-Categorical, ℵ0-stable structures

1985-03-01
Gregory Cherlin , Louise Harrington , A. H. Lachlan

MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

2002-08-02
Zoé Chatzidakis , Ehud Hrushovski , Ya’acov Peterzil
We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the … We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10
Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson
This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and … This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.
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Definable sets over finite fields.

1992-05-01
Zoé Chatzidakis , Lou van den Dries , Angus Macintyre

Introduction to Algebraic Geometry.

1960-06-01
Alice T. Schafer , Serge Lang

Hyperimaginaries and automorphism groups

2001-03-01
Daniel Lascar , Anthony L. Pillay
A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. … A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M) . In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

Strongly minimal countably categorical theories

1981-01-01
Boris Zilber

On groups and fields definable in o-minimal structures

1988-09-01
Anand Pillay

Paires de structures O-minimales

1998-06-01
Yerzhan Baisalov , Bruno Poizat
Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G … Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].

DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01
Mário J. Edmundo , Margarita Otero
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the … Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k&gt;0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
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Dimension of definable sets, algebraic boundedness and Henselian fields

1989-12-01
Lou van den Dries

Stable group theory and approximate subgroups

2011-06-15
Ehud Hrushovski
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math … We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.
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Locally modular theories of finite rank

1986-01-01
Steven Buechler

Generic sets in definably compact groups

2007-01-01
Ya’acov Peterzil , Anand Pillay
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i
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Cell decompositions of C-minimal structures

1994-03-01
Deirdre Haskell , Dugald Macpherson

Algebraic Geometry

1977-01-01
Robin Hartshorne

Topological dynamics and the complexity of strong types

2018-09-08
Krzysztof Krupiński , Anand Pillay , Tomasz Rzepecki
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The geometry of weakly minimal types

1985-12-01
Steven Buechler
Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, … Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p ( M ). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl( X ) ∩ D, Y = acl( Y ) ∩ D and X ∩ Y ≠ ∅, Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp( a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H ∖acl( A ), b ∈ G ∖acl( A ) there are a ′ ∈ H , b ′ ∈ G such that a′ ∈ acl( abb ′ A )∖acl( aA). Similarly when H and G are the realizations of complete types or strong types over A .

Almost orthogonal regular types

1989-12-01
Ehud Hrushovski

On the existence of regular types

1989-12-01
Saharon Shelah , Steven Buechler

Quasi finitely axiomatizable totally categorical theories

1986-01-01
Gisela Ahlbrandt , Martin Ziegler

Model theory and modules

2003-01-01
Mike Prest
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On $ω_1$-categorical theories of fields

1971-01-01
Angus Macintyre
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Bounds in the theory of polynomial rings over fields. A nonstandard approach

1984-02-01
Lou van den Dries , Kai‐Uwe Schmidt

Valued fields, metastable groups

2019-07-23
Ehud Hrushovski , Silvain Rideau
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Definably simple groups in o-minimal structures

2000-02-24
Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

The Mordell-Lang conjecture for function fields

1996-01-01
Ehud Hrushovski
We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in … We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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ON A NONABELIAN BALOG–SZEMERÉDI-TYPE LEMMA

2010-06-08
Tom Sanders
Abstract We show that if G is a group and A ⊂ G is a finite set with ∣ A 2 ∣≤ K ∣ A ∣, then there is a … Abstract We show that if G is a group and A ⊂ G is a finite set with ∣ A 2 ∣≤ K ∣ A ∣, then there is a symmetric neighbourhood of the identity S such that S k ⊂ A 2 A −2 and ∣ S ∣≥exp (− K O ( k ) )∣ A ∣.
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On the category of models of a complete theory

1982-06-01
Daniel Lascar
Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object … Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C ( T ). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C ( T ) determine T ?” There is some obvious limitation: for example let T 0 be the theory of infinite sets (in a language containing only =) and T 1 the theory, in the language ( =, U ( ν 0 ), f ( ν 0 )) stating that: (1) U is infinite. (2) f is a bijective map from U onto its complement. It is quite easy to see that C ( T 0 ) is equivalent to C ( T 1 ). But, in this case, T 0 and T 1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T , defined as follows: (1) The objects are the formulas in the given language. (2) The morphisms from into are the formulas such that (i.e. f defines a map from ϕ into ϕ ; two morphisms defining the same map in all models of T should be identified).