O-Minimality and Diophantine Geometry

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

Publisher: Cambridge University Press

ISBN: 1316301060

Category: Mathematics

Page: N.A

View: 4320

This collection of articles, originating from a short course held at the University of Manchester, explores the ideas behind Pila's proof of the Andre–Oort conjecture for products of modular curves. The basic strategy has three main ingredients: the Pila–Wilkie theorem, bounds on Galois orbits, and functional transcendence results. All of these topics are covered in this volume, making it ideal for researchers wishing to keep up to date with the latest developments in the field. Original papers are combined with background articles in both the number theoretic and model theoretic aspects of the subject. These include Martin Orr's survey of abelian varieties, Christopher Daw's introduction to Shimura varieties, and Jacob Tsimerman's proof via o-minimality of Ax's theorem on the functional case of Schanuel's conjecture.

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

Author: John N. Bray,Derek F. Holt,Colva M. Roney-Dougal

Publisher: Cambridge University Press

ISBN: 1107276225

Category: Mathematics

Page: N.A

View: 2263

This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

Geometric Complex Analysis

In Honor of Kang-Tae Kim’s 60th Birthday, Gyeongju, Korea, 2017

Author: Jisoo Byun,Hong Rae Cho,Sung Yeon Kim,Kang-Hyurk Lee,Jong-Do Park

Publisher: Springer

ISBN: 9811316724

Category: Mathematics

Page: 361

View: 3817

The KSCV Symposium, the Korean Conference on Several Complex Variables, started in 1997 in an effort to promote the study of complex analysis and geometry. Since then, the conference met semi-regularly for about 10 years and then settled on being held biannually. The sixth and tenth conferences were held in 2002 and 2014 as satellite conferences to the Beijing International Congress of Mathematicians (ICM) and the Seoul ICM, respectively. The purpose of the KSCV Symposium is to organize the research talks of many leading scholars in the world, to provide an opportunity for communication, and to promote new researchers in this field.

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

View: 8556

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.

Synthetic Differential Topology

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

Publisher: Cambridge University Press

ISBN: 1108692206

Category: Mathematics

Page: N.A

View: 4471

This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.

A Guide to NIP Theories

Author: Pierre Simon

Publisher: Cambridge University Press

ISBN: 1107057752

Category: Mathematics

Page: 166

View: 5838

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

Asymptotic Analysis in General Relativity

Author: Thierry Daudé,Dietrich Häfner,Jean-Philippe Nicolas

Publisher: Cambridge University Press

ISBN: 1316649407

Category: Mathematics

Page: N.A

View: 9354

Introduction to modern methods for classical and quantum fields in general relativity / Thierry Daudé, Dietrich Häfner, and Jean-Philippe Nicolas -- Geometry of black hole spacetimes / Lars Andersson, Thomas B. Ackdahl, and Pieter Blue -- An introduction to Quantum Field Theory on curved space-times / Christian Gerard -- A minicourse on microlocal analysis for wave propagation / Andras Vasy -- An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity / Sean N. Curry and A. Rod Gover

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

View: 728

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.

Recent Advances in Algebraic Geometry

Author: Christopher D. Hacon,Mircea Mustaţă,Mihnea Popa

Publisher: Cambridge University Press

ISBN: 110764755X

Category: Mathematics

Page: 447

View: 4239

A comprehensive collection of expository articles on cutting-edge topics at the forefront of research in algebraic geometry.

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

View: 7955

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

Finite Structures with Few Types

Author: Gregory L. Cherlin,Ehud Hrushovski

Publisher: Princeton University Press

ISBN: 9780691113319

Category: Mathematics

Page: 193

View: 1037

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.

Cox Rings

Author: Ivan Arzhantsev,Ulrich Derenthal,Jürgen Hausen,Antonio Laface

Publisher: Cambridge University Press

ISBN: 1107024625

Category: Mathematics

Page: 472

View: 6784

This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.

Surveys in Combinatorics 2017

Author: Anders Claesson,Mark Dukes,Sergey Kitaev,David Manlove,Kitty Meeks

Publisher: Cambridge University Press

ISBN: 1108350356

Category: Mathematics

Page: N.A

View: 969

This volume contains nine survey articles which provide expanded accounts of plenary seminars given at the British Combinatorial Conference at the University of Strathclyde in July 2017. This biennial conference is a well-established international event attracting speakers from around the world. Written by internationally recognised experts in the field, these articles represent a timely snapshot of the state of the art in the different areas of combinatorics. Topics covered include the robustness of graph properties, the spt-function of Andrews, switching techniques for edge decompositions of graphs, monotone cellular automata, and applications of relative entropy in additive combinatorics. The book will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.

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

View: 6605

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.

The Arithmetic of Dynamical Systems

Author: J.H. Silverman

Publisher: Springer Science & Business Media

ISBN: 038769904X

Category: Mathematics

Page: 511

View: 1184

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

An Introduction to Diophantine Equations

A Problem-Based Approach

Author: Titu Andreescu,Dorin Andrica,Ion Cucurezeanu

Publisher: Springer Science & Business Media

ISBN: 0817645497

Category: Mathematics

Page: 345

View: 8072

This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.

Model Theory and the Philosophy of Mathematical Practice

Formalization without Foundationalism

Author: John T. Baldwin

Publisher: Cambridge University Press

ISBN: 1108103014

Category: Science

Page: 352

View: 8625

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.

13 Lectures on Fermat's Last Theorem

Author: Paulo Ribenboim

Publisher: Springer Science & Business Media

ISBN: 1468493426

Category: Mathematics

Page: 302

View: 8394

Number Theory

An Introduction to Mathematics

Author: W.A. Coppel

Publisher: Springer Science & Business Media

ISBN: 0387894861

Category: Mathematics

Page: 610

View: 6484

Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included. The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. The prerequisites for this self-contained text are elements from linear algebra. Valuable references for the reader are collected at the end of each chapter. It is suitable as an introduction to higher level mathematics for undergraduates, or for self-study.

Algebraic Informatics

5th International Conference, CAI 2013, Porquerolles, France, September 3-6, 2013. Proceedings

Author: Traian Muntean,Robert Rolland,Dimitrios Poulakis

Publisher: Springer

ISBN: 3642406637

Category: Computers

Page: 275

View: 969

This book constitutes the refereed proceedings of the 5th International Conference on Algebraic Informatics, CAI 2013, held in Porquerolles, France in September 2013. The 19 revised full papers presented together with 5 invited articles were carefully reviewed and selected from numerous submissions. The papers cover topics such as data models and coding theory; fundamental aspects of cryptography and security; algebraic and stochastic models of computing; logic and program modelling.