Part of London Mathematical Society Lecture Note Series
availability: Not yet published - available from July 2016
format: Paperback
isbn: 9781316610442
This in-depth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs. Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. The final four chapters are devoted to the reconstruction problem, and here special emphasis is given to those results that involve the symmetry of graphs, many of which are not to be found in other books. This second edition expands on several of the topics found in the first edition and includes both an enriched bibliography and a wide collection of exercises. Clearer proofs are provided, as are new examples of graphs with interesting symmetry properties. Any student who masters the contents of this book will be well prepared for current research in many aspects of the theory of graph automorphisms and the reconstruction problem.
Preface to the second edition
Preface to the first edition
1. Graphs and groups: preliminaries
2. Various types of graph symmetry
3. Cayley graphs
4. Orbital graphs and strongly regular graphs
5. Graphical regular representations and pseudosimilarity
6. Products of graphs
7. Special classes of vertex-transitive graphs and digraphs
8. The reconstruction conjectures
9. Reconstructing from subdecks
10. Counting arguments in vertex-reconstruction
11. Counting arguments in edge-reconstruction
References
List of notation
Index of terms and definitions.
Part of Cambridge Mathematical Textbooks
availability: Not yet published - available from August 2016
format: Hardback
isbn: 9781107120327
Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since virtually all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.
1. Introduction
2. Basic set building axioms and operations
3. Relations and functions
4. The natural numbers
5. On the size of sets
6. Transfinite recursion
7. The axiom of choice (revisited)
8. Ordinals
9. Cardinals.
Part of Cambridge Studies in Advanced Mathematics
availability: Not yet published - available from July 2016
format: Hardback
isbn: 9781107137240
This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses.
Introduction
Description of the contents
1. Banach space valued martingales
2. Radon Nikodym property
3. Harmonic functions and RNP
4. Analytic functions and ARNP
5. The UMD property for Banach spaces
6. Hilbert transform and UMD Banach spaces
7. Banach space valued H1 and BMO
8. Interpolation methods
9. The strong p-variation of martingales
10. Uniformly convex of martingales
11. Super-reflexivity
12. Interpolation and strong p-variation
13. Martingales and metric spaces
14. Martingales in non-commutative LP *.
Part of London Mathematical Society Lecture Note Series
availability: Not yet published - available from August 2016
format: Paperback
isbn: 9781316613450
D-module theory is essentially the algebraic study of systems of linear partial differential equations. This book, the first devoted specifically to holonomic D-modules, provides a unified treatment of both regular and irregular D-modules. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on D-modules and their tempered holomorphic solutions. As an application, they obtain the Riemann?Hilbert correspondence for regular holonomic D-modules. In the second part of the book the authors do the same for the sheaf of enhanced tempered solutions of (not necessarily regular) holonomic D-modules. Originating from a series of lectures given at the Institut des Hautes Etudes Scientifiques in Paris, this book is addressed to graduate students and researchers familiar with the language of sheaves and D-modules, in the derived sense.
Introduction
1. A review on sheaves and D-modules
2. Indsheaves
3. Tempered solutions of D-modules
4. Regular holonomic D-modules
5. Indsheaves on bordered spaces
6. Enhanced indsheaves
7. Holonomic D-modules
8. Integral transforms
References
List of notations
Index.
Part of Cambridge Tracts in Mathematics
availability: Not yet published - available from August 2016
format: Hardback
isbn: 9781107061576
This unified account of various aspects of a powerful classical method, easy to understand in its simplest forms, is illustrated by applications in several areas of number theory. As well as including diophantine approximation and transcendence, which were mainly responsible for its invention, the author places the method in a broader context by exploring its application in other areas, such as exponential sums and counting problems in both finite fields and the field of rationals. Throughout the book, the method is explained in a 'molecular' fashion, where key ideas are introduced independently. Each application is the most elementary significant example of its kind and appears with detailed references to subsequent developments, making it accessible to advanced undergraduates as well as Master's students in number theory or related areas. It provides 700 exercises both guiding and challenging, while the broad array of applications should interest professionals in fields from number theory to algebraic geometry.
Introduction
1. Prologue
2. Irrationality I
3. Irrationality II - Mahler's method
4. Diophantine equations - Runge's method
5. Irreducibility
6. Elliptic curves - Stepanov's method
7. Exponential sums
8. Irrationality measures I - Mahler
9. Integer-valued entire functions I - Polya
10. Integer-valued entire functions II - Gramain
11. Transcendence I - Mahler
12. Irrationality measures II - Thue
13. Transcendence II - Hermite?Lindemann
14. Heights
15. Equidistribution - Bilu
16. Height lower bounds - Dobrowolski
17. Height upper bounds
18. Counting - Bombieri?Pila
19. Transcendence III - Gelfond?Schneider?Lang
20. Elliptic functions
21. Modular functions
22. Algebraic independence
Appendix: Neron's square root
References
Index.