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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. This book should be of interest to model theorists, analytic geometers and topologists.
Book Synopsis Tame Topology and O-minimal Structures by : Lou Van den Dries
Download or read book Tame Topology and O-minimal Structures written by Lou Van den Dries and published by Cambridge University Press. This book was released on 1998-05-07 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. This book should be of interest to model theorists, analytic geometers and topologists.
Book Synopsis O-minimal Structures by : Mário J. Edmundo
Download or read book O-minimal Structures written by Mário J. Edmundo and published by Cuvillier Verlag. This book was released on 2005 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt:
This book brings the researcher up to date with recent applications of mathematical logic to number theory.
Book Synopsis O-Minimality and Diophantine Geometry by : G. O. Jones
Download or read book O-Minimality and Diophantine Geometry written by G. O. Jones and published by Cambridge University Press. This book was released on 2015-08-13 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings the researcher up to date with recent applications of mathematical logic to number theory.
This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
Book Synopsis Lecture Notes on O-Minimal Structures and Real Analytic Geometry by : Chris Miller
Download or read book Lecture Notes on O-Minimal Structures and Real Analytic Geometry written by Chris Miller and published by Springer Science & Business Media. This book was released on 2012-09-14 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures. In the first part we investigate (for a fixed first-order language L) what we call the L-theory of o-minimality. It is the theory consisting of those L-sentences true in all o-minimal L-structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L-theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list of axioms A which is consistent with the L-theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal. In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting. In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.
Book Synopsis Ultraproducts of O-Minimal Structures by : Alex Rennet
Download or read book Ultraproducts of O-Minimal Structures written by Alex Rennet and published by . This book was released on 2012 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures. In the first part we investigate (for a fixed first-order language L) what we call the L-theory of o-minimality. It is the theory consisting of those L-sentences true in all o-minimal L-structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L-theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list of axioms A which is consistent with the L-theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal. In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting. In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.
This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
Book Synopsis Lecture Notes on O-Minimal Structures and Real Analytic Geometry by : Chris Miller
Download or read book Lecture Notes on O-Minimal Structures and Real Analytic Geometry written by Chris Miller and published by Springer Science & Business Media. This book was released on 2012-09-14 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research in this area.
Book Synopsis Point-Counting and the Zilber–Pink Conjecture by : Jonathan Pila
Download or read book Point-Counting and the Zilber–Pink Conjecture written by Jonathan Pila and published by Cambridge University Press. This book was released on 2022-06-09 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research in this area.
The first book to introduce the rapidly developing subject of NIP theories, for students and researchers in model theory.
Book Synopsis A Guide to NIP Theories by : Pierre Simon
Download or read book A Guide to NIP Theories written by Pierre Simon and published by Cambridge University Press. This book was released on 2015-07-16 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first book to introduce the rapidly developing subject of NIP theories, for students and researchers in model theory.
Leading experts survey the connections between model theory and semialgebraic, subanalytic, p-adic, rigid and diophantine geometry.
Book Synopsis Model Theory, Algebra, and Geometry by : Deirdre Haskell
Download or read book Model Theory, Algebra, and Geometry written by Deirdre Haskell and published by Cambridge University Press. This book was released on 2000-07-03 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leading experts survey the connections between model theory and semialgebraic, subanalytic, p-adic, rigid and diophantine geometry.
These notes give a self-contained treatment of the theory of o-minimal structures.
Book Synopsis Tame Topology and O-Minimal Structures by : Lou Van den Dries
Download or read book Tame Topology and O-Minimal Structures written by Lou Van den Dries and published by . This book was released on 2014-05-14 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes give a self-contained treatment of the theory of o-minimal structures.
