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Model Theory in Algebra, Analysis and Arithmetic - Cetraro, Italy 2012, Editors: H. Dugald Macpherson, Carlo Toffalori (Paperback, 2014 ed.)
Lou Van Den Dries, Jochen Koenigsmann, H. Dugald Macpherson, Anand Pillay, Carlo Toffalori, …
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The book describes 4 main topics in current model theory and
updates their most recent development and applications. The 4
topics are: 1) model theory of valued fields; 2) undecidability in
arithmetic; 3) NIP theories; 4) model theory of real and complex
exponentiation. The book addresses in particular young researchers
in model theory, as well as more senior researchers in other
branches of mathematics.
The book has two chapters. The first chapter is a modern or
contemporary account of stability theory. After a preliminary
section on some of the basic techniques of model theory, the focus
is on local (formula-by-formula) stability theory, treated a little
differently from in the author's Geometric Stability Theory book.
There is also a survey of general and geometric stability theory,
as well as detailed applications to combinatorics (regularity
lemma) using pseudofinite methods.The second chapter is an
introduction to 'continuous logic' or 'continuous model theory',
where truth values are real numbers, drawing on the main texts and
papers, but with an independent point of view. This chapter also
includes some historical background, a discussion of
hyperimaginaries in classical first order model theory, as well as
an introduction to local stability in the continuous framework
making use of some functional analysis results from Grothendieck's
thesis.These chapters are based on notes, written by students, from
a couple of advanced graduate courses in Notre Dame, in Autumn
2018, and Spring 2021.
The book has two chapters. The first chapter is a modern or
contemporary account of stability theory. After a preliminary
section on some of the basic techniques of model theory, the focus
is on local (formula-by-formula) stability theory, treated a little
differently from in the author's Geometric Stability Theory book.
There is also a survey of general and geometric stability theory,
as well as detailed applications to combinatorics (regularity
lemma) using pseudofinite methods.The second chapter is an
introduction to 'continuous logic' or 'continuous model theory',
where truth values are real numbers, drawing on the main texts and
papers, but with an independent point of view. This chapter also
includes some historical background, a discussion of
hyperimaginaries in classical first order model theory, as well as
an introduction to local stability in the continuous framework
making use of some functional analysis results from Grothendieck's
thesis.These chapters are based on notes, written by students, from
a couple of advanced graduate courses in Notre Dame, in Autumn
2018, and Spring 2021.
This introductory treatment covers the basic concepts and machinery
of stability theory. Lemmas, corollaries, proofs, and notes assist
readers in working through and understanding the material and
applications. Full of examples, theorems, propositions, and
problems, it is suitable for graduate students in logic and
mathematics, professional mathematicians, and computer scientists.
Chapter 1 introduces the notions of definable type, heir, and
coheir. A discussion of stability and order follows, along with
definitions of forking that follow the approach of Lascar and
Poizat, plus a consideration of forking and the definability of
types. Subsequent chapters examine superstability, dividing and
ranks, the relation between types and sets of indiscernibles, and
further properties of stable theories. The text concludes with
proofs of the theorems of Morley and Baldwin-Lachlan and an
extension of dimension theory that incorporates orthogonality of
types in addition to regular types.
Since their inception, the Perspectives in Logic and Lecture Notes
in Logic series have published seminal works by leading logicians.
Many of the original books in the series have been unavailable for
years, but they are now in print once again. In this volume, the
fifth publication in the Lecture Notes in Logic series, the authors
give an insightful introduction to the fascinating subject of the
model theory of fields, concentrating on its connections to
stability theory. In the first two chapters David Marker gives an
overview of the model theory of algebraically closed, real closed
and differential fields. In the third chapter Anand Pillay gives a
proof that there are 2 non-isomorphic countable differential closed
fields. Finally, Margit Messmer gives a survey of the model theory
of separably closed fields of characteristic p > 0.
Model theory has made substantial contributions to semialgebraic,
subanalytic, p-adic, rigid and diophantine geometry. These
applications range from a proof of the rationality of certain
Poincare series associated to varieties over p-adic fields, to a
proof of the Mordell-Lang conjecture for function fields in
positive characteristic. In some cases (such as the latter) it is
the most abstract aspects of model theory which are relevant. This
book, originally published in 2000, arising from a series of
introductory lectures for graduate students, provides the necessary
background to understanding both the model theory and the
mathematics behind these applications. The book is unique in that
the whole spectrum of contemporary model theory (stability,
simplicity, o-minimality and variations) is covered and diverse
areas of geometry (algebraic, diophantine, real analytic, p-adic,
and rigid) are introduced and discussed, all by leading experts in
their fields.
The first of a two volume set showcasing current research in model
theory and its connections with number theory, algebraic geometry,
real analytic geometry and differential algebra. Each volume
contains a series of expository essays and research papers around
the subject matter of a Newton Institute Semester on Model Theory
and Applications to Algebra and Analysis. The articles convey
outstanding new research on topics such as model theory and
conjectures around Mordell-Lang; arithmetic of differential
equations, and Galois theory of difference equations; model theory
and complex analytic geometry; o-minimality; model theory and
noncommutative geometry; definable groups of finite dimension;
Hilbert's tenth problem; and Hrushovski constructions. With
contributions from so many leaders in the field, this book will
undoubtedly appeal to all mathematicians with an interest in model
theory and its applications, from graduate students to senior
researchers and from beginners to experts.
The second of a two volume set showcasing current research in model
theory and its connections with number theory, algebraic geometry,
real analytic geometry and differential algebra. Each volume
contains a series of expository essays and research papers around
the subject matter of a Newton Institute Semester on Model Theory
and Applications to Algebra and Analysis. The articles convey
outstanding new research on topics such as model theory and
conjectures around Mordell-Lang; arithmetic of differential
equations, and Galois theory of difference equations; model theory
and complex analytic geometry; o-minimality; model theory and
non-commutative geometry; definable groups of finite dimension;
Hilbert's tenth problem; and Hrushovski constructions. With
contributions from so many leaders in the field, this book will
undoubtedly appeal to all mathematicians with an interest in model
theory and its applications, from graduate students to senior
researchers and from beginners to experts.
Model theory is a branch of mathematical logic that has found applications in several areas of algebra and geometry. It provides a unifying framework for the understanding of old results and more recently has led to significant new results, such as a proof of the Mordell-Lang conjecture for function fields in positive characteristic. Perhaps surprisingly, it is sometimes the most abstract aspects of model theory that are relevant to those applications. This book gives the necessary background for understanding both the model theory and the mathematics behind the applications. Aimed at graduate students and researchers, it contains introductory surveys by leading experts covering the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations), and introducing and discussing the diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) to which the model theory is applied. The book begins with an introduction to model theory by David Marker. It then broadens into three components: pure model theory (Bradd Hart, Dugald Macpherson), geometry(Barry Mazur, Ed Bierstone and Pierre Milman, Jan Denef), and the model theory of fields (Marker, Lou van den Dries, Zoe Chatzidakis).
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.
The model theory of fields is a fascinating subject stretching from
Tarski's work on the decidability of the theories of the real and
complex fields to Hrushovksi's recent proof of the Mordell-Lang
conjecture for function fields. This volume provides an insightful
introduction to this active area, concentrating on connections to
stability theory.
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