April 1, 2026
Paperback ISBN: 9780443339417
Graph theory is a rapidly evolving and expanding mathematical discipline, with new discoveries, challenges, and techniques emerging every year. Graph Theory: Fundamentals and Applications provides a fully up-to-date and accessible introduction to graph theory, covering both the classical and the modern topics, as well as algorithms and evolving challenges addressed by discipline. Based on the latest syllabi and research trends worldwide, this book includes practical, solved problems that are user friendly to undergraduate, postgraduate, and PhD students, and acts as a key aid in learning the fundamentals and the frontiers of graph theory, as well as developing independent problem-solving and critical thinking skills. This book includes clear instruction in graph representation, basic graph operations, graph connectivity, trees and forests, matching theory, planar graphs and graph drawing, algebraic graph theory, graph traversals, network flows, topological graph theory, and cryptography, among other topics. Each chapter features key term definitions, proofs and algorithms, summary points, and unique exercises to reinforce learning, as well as open problems and research challenges that present unsolved or conjectural problems in graph theory for discussion. Supporting student and instructor sites offer additional exercises, solutions, examples, and case studies
1. Introduction to Graph Theory
2. Graph Representation
3. Basic Graph Operations
4. Graph Connectivity
5. Trees and Forests
6. Matching Theory
7. Planar Graphs and Graph Drawing
8. Hamiltonian and Eulerian Graphs
9. Graph Coloring
10. Graph Invariants and Parameters
11. Algebraic Graph Theory
12. Graph Traversals
13. Shortest Path Algorithms
14. Network Flows
15. Topological Graph Theory
16. Ramsey Theory and Extremal Graph Theory
17. Graph Minors and Decompositions
18. Graph Algorithms and Complexity Theory
19. Graphs and Cryptography
20. Graphs and Machine Learning
21. Random Graphs and Probabilistic Methods
22. Research Challenges and Open Problems
23. Appendices
Series: Perspectives in Logic
Published: February 2026
Format: Hardback
ISBN: 9781108490252
Computable structure theory quantifies and studies the relative complexity of mathematical structures. This text, in conjunction with the author's previous volume, represents the first full monograph on computable structure theory in two decades. It brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework. Geared towards graduate students and researchers in mathematical logic, the book enables the reader to learn all the main results and techniques in the area for application in their own research. While the previous volume focused on countable structures whose complexity can be measured within arithmetic, this second volume delves into structures beyond arithmetic, moving into the realm of the hyperarithmetic and the infinitary languages.
Presents the main results and techniques in computable structure theory together in a coherent framework
Makes the subject accessible to graduate students and researchers in logic, from the basic concepts to the frontiers of current research
Includes new work of the author as well as new perspectives on older results
Notation and conventions from computability theory
Notation and conventions from Part I:
1. Ordinals
2. Infinitary logic
3. Computably infinitary languages
4. Pi-one-one sets
5. Hyperarithmetic sets
6. Overspill
7. Forcing
8. The game metatheorem
9. Iterated true-stage arguments
10. Iterating the jump of a structure
11. The isomorphism problem
12. Vaught's conjecture
Bibliography
Index.
Published: April 2026
Format: Hardback
ISBN: 9781009500685
This volume introduces the fundamental results and the state of the art in infinite duration games on graphs. These versatile game models first appeared in the study of automata and logic, but later became important for program verification and synthesis. They have many more applications. In particular, this book covers models from the fields of optimisation, reinforcement learning, model theory, and set theory. The material is presented at a level accessible to beginning graduate students, requiring only a basic understanding of discrete algorithms and graphs. The book's primary objective is constructing efficient algorithms for analysing different types of games. Rather than describe their implementation in full detail, it investigates their theoretical foundations and proves their properties and winning strategies. This tutorial reference from experts across the field is ideal for students, researchers, and professionals in logic and automata, verification and synthesis, and related fields.
Provides a centralized, standalone references for classical and advanced results
Includes state-of-the-art results in all classical models of infinite duration games on graphs
Cements the existing body of knowledge of a dynamic active field
List of contributors
Preface
1. Introduction Nathanaël Fijalkow
I. Classic:
2. Parity games John Fearnley and Nathanaël Fijalkow
3. Regular games Nathanael Fijalkow and Florian Horn
4. Positionality and memory Antonio Casares, Pierre Ohlmann and Pierre Vandenhov
5. Games with payoffs Nathanaël Fijalkow and Benjamin Monmege
II. Stochastic:
6. Markov decision processes Petr Novotný
7. Stochastic games Nathalie Betrand, Patricia Bouyer and Nathanaël Fijalkow
III. Information:
8. Concurrent games Rasmus Ibsen-Jensen
9. Games with signals Hugo Gimbert
10. Synchronous distributed games C. Aiswarya, Paul Gastin and Nathalie Sznajder
IV. Infinite:
11. Timed games Nicolas Markey and Ocan Sankur
12. Pushdown games Arnaud Carayol and Olivier Serre
13. Games with counters Sylvain Schmitz
V. Multi:
14. Games with multiple objectives Mickael Randour
15. Multiplayer games Romain Brenguier and Ocan Sankue
16. Bidding games Guy Avni and Thomas A. Henzinger
Bibliography.
Series: London Mathematical Society Lecture Note Series
Published: May 2026
Format: Paperback
ISBN: 9781009743075
Now in its second edition, this book provides a detailed introduction to the theory of jet bundles. It is written for mathematicians and physicists who wish to study differential equations, particularly those associated with the calculus of variations, in a modern geometric way. A knowledge of differential geometry is assumed, although introductory chapters include the necessary background of fibred manifolds, and on vector and affine bundles. The book explores how first-order jets may be considered as the natural generalisation of vector fields for studying variational problems in field theory, and so many of the constructions are introduced in the context of first- or second-order jets, before being described in their full generality. It features a proof of the local exactness of the variational bicomplex. This edition includes new chapters on velocity bundles and bundles of contact elements, together with updated material on the calculus of variations.
Introduction
1. Bundles
2. Linear bundles
3. Linear operations on general bundles
4. First order jet bundles
5. Second order jet bundles
6. Higher order jet bundles
7. Infinite jet bundles
8. Velocity bundles
9. Jet groups and bundles of contact elements
Bibliography
Glossary of notation
Index.
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Series: New Mathematical Monographs
Published: May 2026
Format: Hardback
ISBN: 9781009617536
Ramification groups of local fields are essential tools for studying boundary behaviour in geometric objects and the degeneration of Galois representations. This book presents a comprehensive development of the recently established theory of upper ramification groups of local fields with imperfect residue fields, starting from the foundations. It also revisits classical theory, including the Hasse–Arf theorem, and offers an optimal generalisation via log monogenic extensions. The conductor of Galois representations, defined through ramification groups, has numerous geometric applications, notably the celebrated Grothendieck–Ogg–Shafarevich formula. A new proof of the Deligne–Kato formula is also provided; this result plays a pivotal role in the theory of characteristic cycles. With a foundational understanding of commutative rings and Galois theory, graduate students and researchers will be well-equipped to engage with this rich area of arithmetic geometry.
A self-contained introduction to the recently established theory of upper ramification groups
Presents a geometric application of the conductor of Galois representations with a new proof
Offers a reference for the theory of log monogenic extension, which serves as a framework for optimal generalisations of the classical theory
Introduction
Part I. Ramification of Henselian Discrete Valuation Fields:
1. Finite extensions
2. Cohomological filtration
Part II. Cyclic Extensions:
3. Cyclic extensions of degree
4. Trace of differential forms
5. The Hasse–Arf theorem
Part III. Conductor and Refinements:
6. Swan conductor
7. Conductor and differential forms
Part IV. Geometric Applications:
8. Grothendieck–Ogg–Shafarevich formula
9. Reduced fiber theorem
10. Nearby cycles on curves
Part V. Upper Ramification Subgroups:
11. Stable integral models
12. Upper ramification subgroups
13. Logarithmic variant and Artin–Schreier–Witt extensions
Part VI. Graded Quotients and Character-Istic Forms:
14. Graded quotients
15. Characteristic forms
16. Logarithmic characteristic forms and the refined Swan con-ductor
Solutions to exercises
References
Index.
Series: New Mathematical Monographs
Published: June 2026
Format: Hardback
ISBN: 9781009388894
This is a contemporary treatment of composition operators on Banach spaces of analytic functions in one complex variable. It provides a step-by-step introduction, starting with a review (including full proofs) of the key tools needed, and building the theory with a focus on Hardy and Bergman spaces. Several proofs of operator boundedness (Littlewood's principle) are given, and the authors discuss approaches to compactness issues and essential norm estimates (Shapiro's theorem) using different tools such as Carleson measures and Nevanlinna counting functions. Membership of composition operators in various ideal classes (Schatten classes for instance) and their singular numbers are studied. This framework is extended to Hardy-Orlicz and Bergman-Orlicz spaces and finally, weighted Hardy spaces are introduced, with a full characterization of those weights for which all composition operators are bounded. This will be a valuable resource for researchers and graduate students working in functional analysis, operator theory, or complex analysis.
Provides step-by-step theory with detailed proofs and examples accessible to both PhD students and advanced researchers
Presents an interdisciplinary approach across operator theory, complex analysis, and Banach spaces
Includes recent, unpublished results in the field, providing a valuable resource for current research and catalyst for further exploration
Foreword
1. Introduction – Toolbox
2. Boundedness of composition operators
3. A first contact with compactness issues
4. Fundamental examples
5. Carleson embedding point of view
6. Compactness via Nevanlinna counting functions and essential norms
7. Carleson versus Nevanlinna
8. Spectrum
9. Schatten classes for composition operators
10. Approximation numbers on Hardy and Bergman spaces
11. Composition operators on Hardy-Orlicz and Bergman-Orlicz spaces
12. Composition operators on weighted Hardy spaces
Appendix A. Schur Test Lemma
Appendix B. Multiplier
Appendix C. Toeplitz operators
Appendix D. Van der Corput and the stationary phase
Appendix E. Several variables
Appendix F. Reminders on subordination of sequences
Appendix G. Singular numbers and their comparisons
Appendix H. Orlicz Spaces
Appendix I. Exercises
Appendix J. A small list of open problems
Bibliography
Index.
Series: Cambridge Studies in Advanced Mathematics
Published: June 2026
Format: Hardback
ISBN: 9781009698559
Principal bundles and their associated fiber bundles famously play a foundational role in both algebraic and differential topology, as well as in fundamental and solid-state physics. More recently, their equivariant and higher homotopy enhancements (gerbes) have been crucial in generalized cohomology theory and for the physics of extended solitons and topological phases. This text is the first to offer a unified perspective of, and introduction to, these topics, providing an insight into material previously scattered across the literature. After a self-contained account of the classical theory of equivariant principal bundles in modern topological groupoid language, the book develops, on the novel backdrop of cohesive higher topos theory, a powerful theory of equivariant principal higher bundles. It establishes new methods like the 'smooth Oka principle' and 'twisted Elmendorf theorem' to elegantly prove classification results and clarify the relation to proper equivariant generalized cohomology theories.
Provides the first accessible account of classical results, presenting equivariant principal bundles in the modern language of topological groupoids
Includes the first coherent account of the novel theory of principal infinity-bundles while generalizing to the equivariant situation
Rederives previous classification results and applications in a streamlined and unified manner, providing new perspectives
What this book is about
Part I. Introduction:
1. Introduction
Part II. In Topological Spaces:
2. Equivariant topology
3. Equivariant principal bundles
Part III. In Cohesive ∞-Stacks:
4. Equivariant ∞-topos theory
5. Equivariant principal ∞-bundles
Part IV. Examples and Applications:
6. Examples and applications
References
Index.
Series: Cambridge Tracts in Mathematics
Published: July 2026
Format: Hardback
ISBN: 9781009687218
Exactly a decade after the publication of the Sz.-Nagy Dilation Theorem, Tsuyoshi Andô proved that, just like for a single contractive operator, every commuting pair of Hilbert-space contractions can be lifted to a commuting isometric pair. Although the inspiration for Andô's proof comes from the elegant construction of Schäffer for the single-variable case, his proof did not shed much light on the explicit nature of the dilation operators and the dilation space as did the original Schäffer and Douglas constructions for a single contraction. Consequently, there has been little follow-up in the direction of a more systematic extension of the Sz.-Nagy–Foias dilation and model theory to the bi-variate setting. Sixty years since the appearance of Andô's first step comes this thorough systematic treatment of a dilation and model theory for pairs of commuting contractions.
Provides a thorough review from scratch of the three proofs of the Sz.-Nagy dilation theorem by Schäffer, Douglas, and Sz.-Nagy–Foias, and of the model theory for isometric pairs by Berger–Coburn–Lebow and Bercovici–Douglas–Foias, including new results
Includes two new elaborate proofs of Andô's Dilation Theorem making use of bi-variate versions of the models of Douglas and Schäffer, respectively
Introduces an appropriate parallel of the Sz.-Nagy–Foias functional model theory for pairs of commuting contractions, including a complete set of unitary invariants for pairs of commuting contractions
Preface
1. Introduction
2. Models for unitary dilations and isometric lifts of a contraction operator
3. The Berger–Coburn–Lebow and Bercovici–Douglas–Foias models for pairs of commuting isometries
4. Andô's dilation and commutant lifting theorems
5. Douglas-type model for Andô isometric lifts
6. Schäffer-type model for Andô isometric lifts
7. Strongly minimal Andô isometric lifts and fundamental operators
8. Pseudo-commuting contractive lifts
9. Functional model and invariants for commuting contractive pairs
Appendix. More general domains and open problems
References
Index.