Geometric Stability Theory

Authors

Anand Pillay

Type: Book

Publication Date: 1996-09-12

Citations: 440

DOI: https://doi.org/10.1093/oso/9780198534372.001.0001

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Abstract

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.

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Geometric stability theory

2018-04-06

Martin Bays

These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that … These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that unimodular (in particular, locally finite or pseudo finite) minimal sets are locally modular; along the way, we discuss the Zilber trichotomy and the group and field confi gurations. We assume the basics of stability theory (forking calculus, U-rank, canonical bases, stable groups and homogeneous spaces), as can be found e.g. in Daniel Palacín's chapter in this volume \[5].

Unidimensional Theories: An introduction to geometric stability theory

1989-01-01

Ehud Hrushovski

Derived categories, stability conditions and geometric applications

2021-09-01

Stability in geometric theories

2004-12-15

Jerry Gagelman

Topological Stability

1994-07-07

Andrew Du Plessis , C. T. C. Wall

This article is an introductory account of work by the authors which is to appear in a book entitled “The geometry of topological stability”. This article is an introductory account of work by the authors which is to appear in a book entitled “The geometry of topological stability”.

Stability, geometry and arithmetic

2019-01-01

Kang Zuo

An introduction to stability theory

2018-04-06

Daniel Palacín

These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical … These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay \[2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler \[4] as an introduction to model theory. Furthermore, the texts of Casanovas \[1] and Wagner \[5] may also be useful to the reader to obtain a different approach to stability theory.

ROBUST DYNAMICS, INVARIANT STRUCTURES AND TOPOLOGICAL CLASSIFICATION

2019-05-01

Rafaël Potrie

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Introduction

1996-09-12

Anand Pillay

Abstract Geometric stability theory is a subject that has grown in an organic fashion out of categoricity theory and classification theory. I will try to describe, in the next few … Abstract Geometric stability theory is a subject that has grown in an organic fashion out of categoricity theory and classification theory. I will try to describe, in the next few paragraphs, and with a minimum of technical language, both the subject and its development, insofar as it is treated in the main body of the book.

Geometric Group Theory

2018-03-28

Cornelia Druţu , Michael Kapovich

Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology … Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology Ultralimits of metric spaces Gromov-hyperbolic spaces and groups Lattices in Lie groups Solvable groups Geometric aspects of solvable groups The Tits alternative Gromov's theorem The Banach-Tarski paradox Amenability and paradoxical decomposition Ultralimits, fixed point properties, proper actions Stallings's theorem and accessibility Proof of Stallings's theorem using harmonic functions Quasiconformal mappings Groups quasiisometric to $\mathbb{H}^n$ Quasiisometries of nonuniform lattices in $\mathbb{H}^n$ A survey of quasiisometric rigidity Appendix: Three theorems on linear groups Bibliography Index

Model theory: an introduction

2003-06-01

David Marker

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, … This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

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Stability Theory and its Variants

2000-01-01

Bradd Hart

Dimension theory plays a crucial technical role in stability theory and its relatives. The abstract dependence relations defined, although combinatorial in nature, often have surprising geometric meaning in particular cases. … Dimension theory plays a crucial technical role in stability theory and its relatives. The abstract dependence relations defined, although combinatorial in nature, often have surprising geometric meaning in particular cases. This article discusses several aspects of dimension theory, such as categoricity, strongly minimal sets, modularity and the Zil’ber principle, forking, simple theories, orthogonality and regular types and in the third, stability, definability of types, stable groups and 1-based groups. One of the achievements of the branch of model theory known as stability theory is the use of numerical invariants, dimensions, in a broad setting. In recent years, this dimension theory has been expanded to include the so-called simple theories. In this paper, I wish to give just a brief overview of the elements of this theory. In the first section, the special case of strongly minimal sets is considered. In the second section, the combinatorial definition of dividing is given and how it leads to a general independence relation is outlined. Only in the third section do stable theories appear and the theory surrounding them is developed there with an eye to other papers in this volume. 1. Strongly Minimal Sets Categorical Theories. One of the simplest questions one can ask about a first order theory is how many models it has of a given cardinality. If T is a countable theory with an infinite model then, by the Lowenheim–Skolem Theorem, it will have at least one model of every infinite power. The situation we will look at first is when a theory has exactly one model of some fixed power. Definition 1.1. A theory T is λ-categorical if T has exactly one model up to isomorphism of cardinality λ. T is said to be totally categorical if T is λ-categorical for every infinite cardinal λ. We will say that T is uncountably categorical if T is λ-categorical for all uncountable λ. Example 1.2. 1. The theory of a set in a language which has only equality is totally categorical.

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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Geometric classification of total stability conditions

2025-02-12

Yu Qiu , Xiaoting Zhang

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.

Algebraic Geometry

2015-12-04

Christopher D. Hacon , Daniel Huybrechts , Yūjirō Kawamata +1 more

The workshop covered a broad variety of areas in algebraic geometry and was the occasion to report on recent advances and works in progress. Special emphasis was put on the … The workshop covered a broad variety of areas in algebraic geometry and was the occasion to report on recent advances and works in progress. Special emphasis was put on the role of derived categories and various stability concepts for sheaves, varieties, complexes, etc. The mix of people working in areas like classification theory, mirror symmetry, derived categories, moduli spaces, p -adic geometry, characteristic p methods, singularity theory led to stimulating discussions.

Simplicity Theory

2013-10-17

Byunghan Kim

This book is about simple first-order theories. The class of simple theories was introduced by S. Shelah in the early 1980s. Then several specific algebraic structures having simple theories have … This book is about simple first-order theories. The class of simple theories was introduced by S. Shelah in the early 1980s. Then several specific algebraic structures having simple theories have been studied by leading researchers, notably by E. Hrushovski. In the mid-1990s the author established in his thesis the symmetry and transitivity of non-forking for simple theories and, with A. Pillay, type-amalgamation for Lascar strong types. Since then a great deal of research work on simplicity theory, the study of simple theories and structures has been produced. This book starts with the introduction of the fundamental notions of dividing and forking, and covers up to the hyperdefinable group configuration theorem for simple theories.

Stable Groups

1997-08-21

Frank Olaf Wagner

The study of stable groups connects model theory, algebraic geometry and group theory. It analyses groups which possess a certain very general dependence relation (Shelah's notion of 'forking'), and tries … The study of stable groups connects model theory, algebraic geometry and group theory. It analyses groups which possess a certain very general dependence relation (Shelah's notion of 'forking'), and tries to derive structural properties from this. These may be group-theoretic (nilpotency or solubility of a given group), algebro-geometric (identification of a group as an algebraic group), or model-theoretic (description of the definable sets). In this book, the general theory of stable groups is developed from the beginning (including a chapter on preliminaries in group theory and model theory), concentrating on the model- and group-theoretic aspects. It brings together the various extensions of the original finite rank theory under a unified perspective and provides a coherent exposition of the knowledge in the field.

Stability theorems

2003-09-11

F. E. A. Johnson

In this chapter we prove three of the six theorems stated in the Introduction. We have seen that stabilization induces a tree structure on the stable module Ω3(Z). Two-dimensional homotopy … In this chapter we prove three of the six theorems stated in the Introduction. We have seen that stabilization induces a tree structure on the stable module Ω3(Z). Two-dimensional homotopy types possess an analogous tree structure; homotopy types at the bottom level are said to be minimal. We say that G has the realization property when all algebraic 2-complexes over Z[G] are geometrically realizable. The first of our results is then:

Analytic and pseudo-analytic structures

2017-04-26

Boris Zilber

One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly thatmore andmore ofmodel theoretic research focuses … One of the questions frequently asked nowadays about model theory is whether it is still logic. The reason for asking the question is mainly thatmore andmore ofmodel theoretic research focuses on concretemathematical fields, uses extensively their tools and attacks their inner problems. Nevertheless the logical roots in the case of model theoretic geometric stability theory are not only clear but also remain very important in all its applications.

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Forking in pregeometries.

1998-01-01

Rami Grossberg , Olivier Lessmann

The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric … The aim of this paper is to set the foundation to separate geometric model theory from model theory. Our thesis is that it is possible to lift results from geometric model theory to non first order logic (e.g. LWl)W). We introduce a relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This is important when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].

Expansions géométriques et ampleur

2015-06-10

Juan Felipe Carmona

Le resultat principal de cette these est l'etude de l'ampleur dans des expansions des structures geometriques et de SU-rang omega par un predicat dense/codense independant. De plus, nous etudions le … Le resultat principal de cette these est l'etude de l'ampleur dans des expansions des structures geometriques et de SU-rang omega par un predicat dense/codense independant. De plus, nous etudions le rapport entre l'ampleur et l'equationalite, donnant une preuve directe de l'equationalite de certaines theories CM-triviales. Enfin, nous considerons la topologie indiscernable et son lien avec l'equationalite et calculons la complexite indiscernable du pseudoplan libre

Model-theoretic constructions via amalgamation and reducts

2007-01-01

David M. Evans

Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples … Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples are theories of vector spaces and algebraically closed fields; also the degenerate example of the theory of infinite ‘pure’ sets where the only structure comes from equality. Algebraic closure in a strongly minimal structure satisfies the exchange condition, so gives rise to notions of dimension and independence (corresponding to linear dimension/ independence and transcendence degree/ algebraic independence in the two classical examples).

Homogeneity in the free group

2012-10-01

Chloé Perin , Rizos Sklinos

We show that any nonabelian free group F is strongly ℵ0-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under … We show that any nonabelian free group F is strongly ℵ0-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under Aut(F). We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly ℵ0-homogeneous.

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Functoriality and uniformity in Hrushovski's groupoid-cover correspondence

2018-03-30

Levon Haykazyan , Rahim Moosa

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Eberlein oligomorphic groups

2017-03-16

Itaï Ben Yaacov , Tomás Ibarlucía , Todor Tsankov

We study the Fourier–Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups … We study the Fourier–Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier–Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub><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:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:annotation encoding="application/x-tex">\aleph _0</mml:annotation></mml:semantics></mml:math></inline-formula>-categorical structures. This analysis is then extended to all semitopological semigroup compactifications<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>of such a group:<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"><mml:semantics><mml:mi>S</mml:mi><mml:annotation encoding="application/x-tex">S</mml:annotation></mml:semantics></mml:math></inline-formula>is Hilbert-representable if and only if it is an inverse semigroup. We also show that every factor of the Hilbert compactification is Hilbert-representable.

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None

2002-01-01

Vera Puninskaya

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Infinite stable graphs with large chromatic number

2021-12-08

Yatir Halevi , Itay Kaplan , Saharon Shelah

We prove that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals left-parenthesis upper V comma upper E right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G=(V,E)</mml:annotation></mml:semantics></mml:math></inline-formula>is an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable (respectively, superstable) graph with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … We prove that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals left-parenthesis upper V comma upper E right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G=(V,E)</mml:annotation></mml:semantics></mml:math></inline-formula>is an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable (respectively, superstable) graph with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi left-parenthesis upper G right-parenthesis greater-than normal alef 0"><mml:semantics><mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">\chi (G)>\aleph _0</mml:annotation></mml:semantics></mml:math></inline-formula>(respectively,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript normal alef 0"><mml:semantics><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">2^{\aleph _0}</mml:annotation></mml:semantics></mml:math></inline-formula>) then<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>contains all the finite subgraphs of the shift graph<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal h Subscript n Baseline left-parenthesis omega right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">h</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Sh}_n(\omega )</mml:annotation></mml:semantics></mml:math></inline-formula>for some<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if<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="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper U left-parenthesis upper G right-parenthesis less-than-or-equal-to 2"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">U</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\operatorname {U}(G)\leq 2</mml:annotation></mml:semantics></mml:math></inline-formula>we prove that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 2"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n\leq 2</mml:annotation></mml:semantics></mml:math></inline-formula>suffices.

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On function field Mordell–Lang and Manin–Mumford

2016-03-07

Franck Benoist , Élisabeth Bouscaren , Anand Pillay

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the … We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

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Types on stable Banach spaces

1998-01-01

José Iovino

We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\widehat{X}$ is an … We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\widehat{X}$ is an ultrapower of X and B is a ball in $\widehat{X}$, the intersection B ∩ X c

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SOME DEFINABLE GALOIS THEORY AND EXAMPLES

2017-06-01

Omar León Sánchez , Anand Pillay

Abstract We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois … Abstract We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics , 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ -groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics , 281(1), 2016].

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Saturated free algebras and almost indiscernible theories

2022-02-01

T. G. Kucera , Anand Pillay

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THE FIELDS OF REAL AND COMPLEX NUMBERS WITH A SMALL MULTIPLICATIVE GROUP

2006-06-09

Lou van den Dries , Ayhan Günaydın

We consider the model theory of the real and complex fields with a multiplicative group having the Mann property. Among these groups are the finitely generated multiplicative groups in these … We consider the model theory of the real and complex fields with a multiplicative group having the Mann property. Among these groups are the finitely generated multiplicative groups in these fields. As a by-product we obtain some results on groups with the Mann property in rings of Witt vectors and in fields of positive characteristic.k

Stability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaces

2020-07-01

Karim Khanaki

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain … We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $\phi(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $\phi$-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-\v{S}mulian theorem.

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Hodge Theory for Combinatorial Geometries

2015-11-09

Karim Adiprasito , June Huh , Eric Katz

We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of … We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.

A group version of stable regularity

2018-10-24

Gabriel Conant , Anthony L. Pillay , C. Terry

Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A … Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k -stable. Then there is a normal subgroup H ≤ G of index at most n , and a set Y ⊆ G , which is a union of cosets of H , such that | A △ Y | ≤ε| H |. It follows that, for any coset C of H , either | C ∩ A |≤ ε| H | or | C \ A | ≤ ε | H |. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$ .

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Generic automorphisms and green fields

2011-12-06

Martin Hils

We show that the generic automorphism is axiomatizable in the green field of Poizat (once Morleyized) as well as in the bad fields that are obtained by collapsing this green … We show that the generic automorphism is axiomatizable in the green field of Poizat (once Morleyized) as well as in the bad fields that are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain ‘bad pseudofinite fields’ in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatization. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, for example, the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.

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Topological dynamics and definable groups

2013-05-15

Anand Pillay

Abstract We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G ( … Abstract We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G ( M ) on its “external type space” S G.ext ( M ), can explain, account for, or give rise to, the quotient G / G 00 , at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or f sg ) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of G ( M ) for measure-stable groups. We also point out that for G definably amenable (in a NIP theory) G / G 00 can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global f -generic types.

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On NIP and invariant measures

2007-01-01

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 math.LO/0607442. Among key results are: (i) if … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ 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 $bdd(A)$ (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 translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.

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Forking in the free group

2007-02-06

Anand Pillay

We study model-theoretic and stability-theoretic properies of the nonabelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn. We point out … We study model-theoretic and stability-theoretic properies of the nonabelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn. We point out analogies between the free group and so-called bad groups of finite Morley rank and we prove non CM-triviality of the free group.

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On compactifications and the topological dynamics of definable groups

2012-12-13

Jakub Gismatullin , Davide Penazzi , Anand Pillay

We discuss compactifications and topological dynamics. For G a group in some structure M, we define notions of definable compactification of G and definable action of G on a compact … We discuss compactifications and topological dynamics. For G a group in some structure M, we define notions of definable compactification of G and definable action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the compactification of G as G*/G*00_M and the G-ambit as the type space S_{G}(M). We also prove existence and uniqueness of universal minimal G-flows, and discuss issues of amenability and extreme amenability in this category, with a characterization of the latter. For the sake of completeness we also describe the (Bohr) compactification and G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).

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The stable regularity lemma revisited

2015-04-23

M. Malliaris , Anand Pillay

We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and … We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability.

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On stable fields and weight

2010-07-01

Krzysztof Krupiński , Anand Pillay

Abstract We prove that if K is an (infinite) stable field whose generic type has weight 1, then K is separably closed. We also obtain some partial results about stable … Abstract We prove that if K is an (infinite) stable field whose generic type has weight 1, then K is separably closed. We also obtain some partial results about stable groups and fields whose generic type has finite weight, as well as about strongly stable fields (where by definition all types have finite weight).

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On the binding group in simple theories

2002-09-01

Ziv Shami , Frank Olaf Wagner

Abstract We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p ′ interalgebraic … Abstract We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p ′ interalgebraic with a finite tuple of realizations of p , which is generated over φ . Moreover, the group of elementary permutations of p ′ over all realizations of φ is type-definable.

Co-theory of sorted profinite groups for PAC structures

2022-11-09

Daniel Max Hoffmann , Junguk Lee

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding … We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOP n hierarchy. Fifth, we characterize the algebraic closure in PAC structures.

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Groups definable in two orthogonal sorts

2015-09-01

Alessandro Berarducci , Marcello Mamino

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Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields

2016-01-01

Vahagn Aslanyan

Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the … Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ where $t$ is an element with $t'=1$. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U$. Moreover, the induced structure on Cartesian powers of $U$ is given by special subvarieties. If $E$ has some special form then all fibres $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (with $s$ non-constant) have the same properties. In particular, since the $j$-function satisfies an Ax-Schanuel theorem of the required form (due to Pila and Tsimerman), our results will give another proof for a theorem of Freitag and Scanlon stating that the differential equation of $j$ defines a strongly minimal set with trivial geometry (which is not $\aleph_0$-categorical though).

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INTERPRETING GROUPS AND FIELDS IN SOME NONELEMENTARY CLASSES

2005-06-01

Tapani Hyttinen , Olivier Lessmann , Saharon Shelah

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem. Let [Formula: see text] be a large homogeneous model of … This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ S D (A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a 1 , …, a n ∈ P and finite subset C ⊆ Q, but [Formula: see text] for some independent a 1 , …, a n , a n+1 ∈ P and some finite subset C ⊆ Q. Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and •If n = 1 then G is abelian and acts regularly on P′. •If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K. •If n = 3 the action of G on P′ is isomorphic to the action of PGL 2 (K) on the projective line ℙ 1 (K) of the algebraically closed field K. We prove a similar result for excellent classes.

On the canonical base property

2012-05-27

Ehud Hrushovski , Daniel Palacín , Anand Pillay

We give an example of a finite rank, in fact aleph-1 categorical theory where the CBP (canonical base property) does not hold. We include a group-like example. We also prove, … We give an example of a finite rank, in fact aleph-1 categorical theory where the CBP (canonical base property) does not hold. We include a group-like example. We also prove, in a finite Morley rank context, if all definable Galois groups are rigid then the theory has the CBP (in a strong sense).

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Definable subgroups of measure algebras

2006-07-24

Alexander Berenstein

Abstract We show that type‐definable subgroups of measure algebras are definable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) Abstract We show that type‐definable subgroups of measure algebras are definable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Logical perfection in mathematics and beyond

2018-03-13

John Alexander Cruz Morales , Boris Zilber

In this paper we discuss the notion of logically perfect structures in mathematics, introduce by the second author, and argue that those structures might also be of interest in physics. In this paper we discuss the notion of logically perfect structures in mathematics, introduce by the second author, and argue that those structures might also be of interest in physics.

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Analytic and pseudo-analytic structures (a survey)

2004-01-01

Boris Zilber

The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the … The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the Trichotomy conjecture.

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Imaginaries in equicharacteristic zero henselian fields

2023-01-01

Silvain Rideau , Mariana Vicaría

We prove an elimination of imaginaires results for (almost all) henselian valued fields of equicharacteristic zero. To do so, we consider a mix of sorts introduced in earlier works of … We prove an elimination of imaginaires results for (almost all) henselian valued fields of equicharacteristic zero. To do so, we consider a mix of sorts introduced in earlier works of the two authors and define a generalized version of the k-linear imaginaries. For a large class of value groups containing all subgroups of (\mathbb{R}^n) for some (n), we prove that the imaginaries of such a valued field can be elimininated in the field, the k-linear imaginaries and the imaginaries of the value group.

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Model-theoretic Elekes–Szabó for stable and o-minimal hypergraphs

2024-02-15

Artem Chernikov , Ya’acov Peterzil , Sergei Starchenko

A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity … A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension definable in: (1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic 0) and (2) o-minimal expansions of groups. Our methods provide explicit bounds on the power-saving exponent in the nongroup case. Ingredients of the proof include a higher arity generalization of the abelian group configuration theorem in stable structures (along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups) and Zarankiewicz-style bounds for hypergraphs definable in distal structures.

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Definability and Invariance

2007-06-01

Newton C. A. da Costa , Alexandre A. M. Rodrigues

On the generic type of the free group

2011-01-04

Rizos Sklinos

We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_{\omega}$. We also prove that the … We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_{\omega}$. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not $\aleph_1$-homogeneous.

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A note on stable sets, groups, and theories with NIP

2007-05-18

Alf Onshuus , Ya’acov Peterzil

Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x … Abstract Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x ) ∧ α ( x , y ) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G / H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Zilber’s restricted trichotomy in characteristic zero

2023-09-27

Benjamin Castle

We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<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>is any non-locally modular … We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<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>is any non-locally modular strongly minimal structure interpreted in an algebraically closed field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>of characteristic zero, then<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>itself interprets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>; in particular, any non-1-based structure interpreted in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>is mutually interpretable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.

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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THE GROUP CONFIGURATION THEOREM FOR GENERICALLY STABLE TYPES

2024-11-11

Paul Wang

Abstract We generalize Hrushovski’s group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically … Abstract We generalize Hrushovski’s group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically stable types, which we recall in the second section, enable us to adapt the proof known in the stable context.

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Locally modular regular types

1987-01-01

Ehud Hrushovski

Unimodular Minimal Structures

1992-12-01

Ehud Hrushovski

Lectures in projective geometry

1962-01-01

A. Seidenberg

Zariski Geometries

2010-02-01

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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