O-Minimality and Diophantine Geometry

Author: A. J. Wilkie,G. O. Jones

Publisher: Cambridge University Press

ISBN: 1107462495

Category: Mathematics

Page: 232

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Brings the researcher up to date with recent applications of mathematical logic to number theory.

Algebraic Geometry

Salt Lake City 2015 : 2015 Summer Research Institute, July 13-31, 2015, University of Utah, Salt Lake City, Utah

Author: Richard Thomas

Publisher: American Mathematical Soc.

ISBN: 1470435780

Category: Geometry, Algebraic

Page: 635

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This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Arithmetic and Geometry

Author: Luis Dieulefait,Gerd Faltings,D. R. Heath-Brown,Yuri I. Manin,B. Z. Moroz,Yu. V. Manin,Jean-Pierre Wintenberger

Publisher: Cambridge University Press

ISBN: 1107462541

Category: Mathematics

Page: 550

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The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.

Synthetic Differential Topology

Author: Marta Bunge,Felipe Gago,Ana María San Luis

Publisher: Cambridge University Press

ISBN: 1108447236

Category: Mathematics

Page: N.A

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Represents the state of the art in the new field of synthetic differential topology.

Prospects in Mathematics

Author: Friedrich Ernst Peter Hirzebruch,Friedrich Hirzebruch,Lars Hörmander,John Milnor,Jean-Pierre Serre,Et Al,I. M. Singer,John N. Mather,Friedrich Serre,I M Hormander,Phillip Griffiths

Publisher: Princeton University Press

ISBN: 9780691080949

Category: Mathematics

Page: 185

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Prospects in Mathematics, a symposium, was held in Princeton on March 16-18, 1970 on the occasion of the dedication of the university's new Fine Hall. Invited addresses were delivered by F.E.P. Hirzebruch, L. Hormander, J. Milnor, J.-P. Serre, and I.M. Singer. The lecturers undertook to describe some current trends in mathematics in the prespective of the recent past and in terms of some of the expectations for the future. Among subjects discussed were algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations. This volume presents the written texts of those lectures. (Author).

Finite Structures with Few Types

Author: Gregory L. Cherlin,Ehud Hrushovski

Publisher: Princeton University Press

ISBN: 9780691113319

Category: Mathematics

Page: 193

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This book applies model theoretic methods to the study of certain finite permutation groups, the automorphism groups of structures for a fixed finite language with a bounded number of orbits on 4-tuples. Primitive permutation groups of this type have been classified by Kantor, Liebeck, and Macpherson, using the classification of the finite simple groups. Building on this work, Gregory Cherlin and Ehud Hrushovski here treat the general case by developing analogs of the model theoretic methods of geometric stability theory. The work lies at the juncture of permutation group theory, model theory, classical geometries, and combinatorics. The principal results are finite theorems, an associated analysis of computational issues, and an "intrinsic" characterization of the permutation groups (or finite structures) under consideration. The main finiteness theorem shows that the structures under consideration fall naturally into finitely many families, with each family parametrized by finitely many numerical invariants (dimensions of associated coordinating geometries). The authors provide a case study in the extension of methods of stable model theory to a nonstable context, related to work on Shelah's "simple theories." They also generalize Lachlan's results on stable homogeneous structures for finite relational languages, solving problems of effectivity left open by that case. Their methods involve the analysis of groups interpretable in these structures, an analog of Zilber's envelopes, and the combinatorics of the underlying geometries. Taking geometric stability theory into new territory, this book is for mathematicians interested in model theory and group theory.

A Guide to NIP Theories

Author: Pierre Simon

Publisher: Cambridge University Press

ISBN: 1107057752

Category: Mathematics

Page: 166

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The first book to introduce the rapidly developing subject of NIP theories, for students and researchers in model theory.

The Birth of Model Theory

Lowenheim's Theorem in the Frame of the Theory of Relatives

Author: Calixto Badesa

Publisher: Princeton University Press

ISBN: 9781400826186

Category: Mathematics

Page: 256

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Löwenheim's theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory--that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with Löwenheim's theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem--a logician raised in the algebraic tradition, like Löwenheim--appears to have attributed to him. In The Birth of Model Theory, Calixto Badesa provides both the first sustained, book-length analysis of Löwenheim's proof and a detailed description of the theoretical framework--and, in particular, of the algebraic tradition--that made the theorem possible. Badesa's three main conclusions amount to a completely new interpretation of the proof, one that sharply contradicts the core of modern scholarship on the topic. First, Löwenheim did not use an infinitary language to prove his theorem; second, the functional interpretation of Löwenheim's normal form is anachronistic, and inappropriate for reconstructing the proof; and third, Löwenheim did not aim to prove the theorem's weakest version but the stronger version Skolem attributed to him. This book will be of considerable interest to historians of logic, logicians, philosophers of logic, and philosophers of mathematics.

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

Author: Ehud Hrushovski,François Loeser

Publisher: Princeton University Press

ISBN: 1400881226

Category: Mathematics

Page: 232

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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.

Model Theory of Fields

Author: David Marker,Margit Messmer,Anand Pillay

Publisher: Cambridge University Press

ISBN: 1107168074

Category: Mathematics

Page: 164

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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 2א non-isomorphic countable differential closed fields. Finally, Margit Messmer gives a survey of the model theory of separably closed fields of characteristic p > 0.

Categoricity

Author: John T. Baldwin

Publisher: American Mathematical Soc.

ISBN: 0821848933

Category: Mathematics

Page: 235

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Modern model theory began with Morley's categoricity theorem: A countable first-order theory that has a unique (up to isomorphism) model in one uncountable cardinal (i.e., is categorical in cardinality) if and only if the same holds in all uncountable cardinals. Over the last 35 years Shelah made great strides in extending this result to infinitary logic, where the basic tool of compactness fails. He invented the notion of an Abstract Elementary Class to give a unifying semantic account of theories in first-order, infinitary logic and with some generalized quantifiers. Zilber developed similar techniques of infinitary model theory to study complex exponentiation. This book provides the first unified and systematic exposition of this work. The many examples stretch from pure model theory to module theory and covers of Abelian varieties. Assuming only a first course in model theory, the book expounds eventual categoricity results (for classes with amalgamation) and categoricity in excellent classes. Such crucial tools as Ehrenfeucht - Mostowski models, Galois types, tameness, omitting-types theorems, multi-dimensional amalgamation, atomic types, good sets, weak diamonds, and excellent classes are developed completely and methodically. The (occasional) reliance on extensions of basic set theory is clearly laid out. The book concludes with a set of open problems.

Around the Zilber-Pink Conjecture

Author: Philipp Habegger,Gaël Rémond,Thomas Scanlon,Andrei Yafaev,Emmanuel Ullmo

Publisher: N.A

ISBN: 9782856298565

Category: Arithmetical algebraic geometry

Page: 284

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"Following Faltings and Vojta's work proving the Mordell-Lang conjecture for abelian varieties and Raynaud's work proving the Manin-Mumford conjecture, many new diophantine questions appeared, often described as problems of unlikely intersections. The arithmetic of moduli spaces of abelian varieties and, more generally, Shimura varieties has been parallel-developed around the central André-Oort conjecture. These two themes can be placed in a common frame--the Zilber-Pink conjecture. This volume is an introduction to these problems and to the various techniques used: geometry, height theory, reductive groups and Hodge theory, Shimura varieties, and model theory via the notion of o-minimal structure."- publisher

Progress and Challenges in Dynamical Systems

Proceedings of the International Conference Dynamical Systems: 100 Years after Poincaré, September 2012, Gijón, Spain

Author: Santiago Ibáñez,Jesús S. Pérez del Río,Antonio Pumariño,J. Ángel Rodríguez

Publisher: Springer Science & Business Media

ISBN: 3642388302

Category: Mathematics

Page: 411

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This book contains papers based on talks given at the International Conference Dynamical Systems: 100 years after Poincaré held at the University of Oviedo, Gijón in Spain, September 2012. It provides an overview of the state of the art in the study of dynamical systems. This book covers a broad range of topics, focusing on discrete and continuous dynamical systems, bifurcation theory, celestial mechanics, delay difference and differential equations, Hamiltonian systems and also the classic challenges in planar vector fields. It also details recent advances and new trends in the field, including applications to a wide range of disciplines such as biology, chemistry, physics and economics. The memory of Henri Poincaré, who laid the foundations of the subject, inspired this exploration of dynamical systems. In honor of this remarkable mathematician, theoretical physicist, engineer and philosopher, the authors have made a special effort to place the reader at the frontiers of current knowledge in the discipline.

Handbook of Dynamical Systems

Volume 1B

Author: A. Katok,B. Hasselblatt

Publisher: Elsevier

ISBN: 9780080478227

Category: Mathematics

Page: 1234

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This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey “Principal Structures of Volume 1A. The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field. . The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.

Introduction to the Theory of Schemes

Author: Yuri I. Manin

Publisher: Springer

ISBN: 3319743163

Category: Mathematics

Page: 205

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This English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas. "This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander Beilinson

Introduction to Arithmetic Groups

Author: Dave Witte Morris

Publisher: N.A

ISBN: 9780986571602

Category: Mathematics

Page: 492

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This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n, Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background. A PDF file of the book is available on the internet. This inexpensive printed edition is for readers who prefer a hardcopy.

Mathematics of Public Key Cryptography

Author: Steven D. Galbraith

Publisher: Cambridge University Press

ISBN: 1107013925

Category: Computers

Page: 615

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This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography.

A Field Guide to Algebra

Author: Antoine Chambert-Loir

Publisher: Springer Science & Business Media

ISBN: 038726955X

Category: Mathematics

Page: 198

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This book has a nonstandard choice of topics, including material on differential galois groups and proofs of the transcendence of e and pi. The author uses a conversational tone and has included a selection of stamps to accompany the text.

Complex Dynamics

Author: Lennart Carleson,Theodore W. Gamelin

Publisher: Springer Science & Business Media

ISBN: 1461243645

Category: Mathematics

Page: 192

View: 1949

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A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.