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One dimensional structures in o-minimal theories

Authors

Assaf Hasson, Alf Onshuus, Ya’acov Peterzil
Type: Preprint
Publication Date: 2007-01-01
Citations: 2
DOI: https://doi.org/10.48550/arxiv.math/0702748

Abstract

We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The main tool is a theory for intersection of curves which we develop.

Locations

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Definable structures in o-minimal theories: One dimensional types

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil
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Definable one dimensional structures in o-minimal theories

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

One-dimensional groups definable in o-minimal structures

1994-10-01
Adam Strzeboński

Higher Dimensional Trichotomy Conjectures in Model Theory

2021-01-01
Benjamin Castle
Author(s): Castle, Benjamin | Advisor(s): Scanlon, Thomas | Abstract: In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of … Author(s): Castle, Benjamin | Advisor(s): Scanlon, Thomas | Abstract: In this thesis we study the Restricted Trichotomy Conjectures for algebraically closed and o-minimal fields. These conjectures predict a classification of all sufficiently complex, that is, non-locally modular, strongly minimal structures which can be interpreted from such fields. Such problems have been historically divided into `lower dimensional' and `higher dimensional' cases; this thesis is devoted to a number of partial results in the higher dimensional cases. In particular, in ACF_0 and over o-minimal fields, we prove that all higher dimensional strongly minimal structures whose definable sets satisfy certain geometric restrictions are locally modular. We also make progress toward verifying these geometric restrictions in any counterexample. Finally, in the last chapter we give a full proof of local modularity for strongly minimal expansions of higher dimensional groups in ACF_0.

Zil’ber’s Trichotomy and o-minimal Structures

1998-01-01
Ya’acov Peterzil
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Zil'ber's Trichotomy and o-minimal Structures

2017-03-02
Ya’acov Peterzil
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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On lovely pairs of geometric structures

2009-11-29
Alexander Berenstein , Evgueni Vassiliev

Intersection theory for o-minimal manifolds

2001-01-01
Alessandro Berarducci , Margarita Otero

The O-minimal Zilber Conjecture in Higher Dimensions

2024-06-13
Benjamin Castle
We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let … We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let $M$ be an interpretable set in $\mathcal R$, and let $\mathcal M=(M,...)$ be a reduct of the induced structure on $M$. If $\mathcal M$ is strongly minimal and not locally modular, then $\dim_{\mathcal R}(M)=2$. As an application, we prove the Zilber trichotomy for all strongly minimal structures interpreted in the theory of compact complex manifolds.
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Restricted trichotomy in higher dimensions

2016-01-01
Dmitry Sustretov
I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic … I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic relations definable in the Zariski language on $M$, then $M$ is locally modular.
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Restricted trichotomy in higher dimensions

2016-04-19
Dmitry Sustretov
I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic … I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic relations definable in the Zariski language on $M$, then $M$ is locally modular.

Tame Topology and O-minimal Structures

1998-05-07
L. P. D. van den Dries
Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment … Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

Mini-Workshop: Topological and Differential Expansions of o-minimal Structures

2023-07-26
Paola D’Aquino , Pantelis E. Eleftheriou , Omar León Sánchez +1 more
The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame … The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame geometric properties to expansions of specified fields by classical operators that preserve neo-stability properties. There were presentations and discussions on recent developments in definable groups and decompositions in relatively tame setups, the interplay of different notions of dimension and closure operators, and applications of the model theory of differential fields to diophantine geometry.

Definable one dimensional topologies in o-minimal structures

2018-07-21
Ya’acov Peterzil , Ayala Rosel
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.

Definable one dimensional topologies in o-minimal structures

2018-01-01
Ya’acov Peterzil , Ayala Rosel
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.
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SHEAF COHOMOLOGY IN o-MINIMAL STRUCTURES

2006-12-01
Mário J. Edmundo , G. O. Jones , Nicholas J. Peatfield
Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures. Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.
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Covers of groups definable in o-minimal structures

2005-01-01
Mário J. Edmundo
In this paper we develop the theory of covers for locally definable groups in o-minimal structures. In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
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A theorem on generic intersections in an o-minimal structure

2014-01-01
Krzysztof Jan Nowak
Consider a transitive definable action of a Lie group $G$ on a definable manifold $M$. Given two (locally) definable subsets $A$ and $B$ of $M$, we prove that the dimension … Consider a transitive definable action of a Lie group $G$ on a definable manifold $M$. Given two (locally) definable subsets $A$ and $B$ of $M$, we prove that the dimension of the intersection $\sigma (A) \cap B$ is not greater than the expected one for a
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Definable one-dimensional topologies in O-minimal structures

2019-05-31
Ya’acov Peterzil , Ayala Rosel
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O-minimality and certain atypical intersections

2014-01-01
Philipp Habegger , Jonathan Pila
We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We … We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.
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Definable structures in o-minimal theories: One dimensional types

2010-10-31
Assaf Hasson , Alf Onshuus , Ya’acov Peterzil
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Unstable structures definable in o-minimal theories

2010-03-09
Assaf Hasson , Alf Onshuus
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