Part of Cambridge Studies in Advanced Mathematics
available from December 2024
format: Hardback isbn: 9781009449465
Offering a unique resource for advanced graduate students and researchers, this book treats the fundamentals of Quillen model structures on abelian and exact categories. Building the subject from the ground up using cotorsion pairs, it develops the special properties enjoyed by the homotopy category of such abelian model structures. A central result is that the homotopy category of any abelian model structure is triangulated and characterized by a suitable universal property ? it is the triangulated localization with respect to the class of trivial objects. The book also treats derived functors and monoidal model categories from this perspective, showing how to construct tensor triangulated categories from cotorsion pairs. For researchers and graduate students in algebra, topology, representation theory, and category theory, this book offers clear explanations of difficult model category methods that are increasingly being used in contemporary research.
Clearly explains abstract concepts including an abundance of diagrammatic proofs
Uses an accessible approach, developing the fundamentals of abelian model categories from the cotorsion pair perspective
Develops theory within the more general setting of Quillen exact categories, making results more widely applicable
Introduction and main examples:
1. Additive and exact categories
2. Cotorsion pairs
3. Stable categories from cotorsion pairs
4. Hovey triples and abelian model structures
5. The homotopy category of an abelian model structure
6. The triangulated homotopy category
7. Derived functors and abelian monoidal model structures
8. Hereditary model structures
9. Constructing complete cotorsion pairs
10. Abelian model structures on chain complexes
11. Mixed model structures and examples
12. Cofibrant generation and well-generated homotopy categories
A. Hovey's correspondence for general exact categories
B. Right and left homotopy relations
C. Bibliographical notes
References
Index.
Part of Elements in Non-local Data Interactions: Foundations and Applications
available from January 2025
format: Paperback isbn: 9781009346634
The use of differential equations on graphs as a framework for the mathematical analysis of images emerged about fifteen years ago and since then it has burgeoned, and with applications also to machine learning. The authors have written a bird's eye view of theoretical developments that will enable newcomers to quickly get a flavour of key results and ideas. Additionally, they provide an substantial bibliography which will point readers to where fuller details and other directions can be explored. This title is also available as open access on Cambridge Core.
1. Introduction
2. History and literature overview
3. Calculus on undirected edge-weighted graphs
4. Directed graphs
5. The graph Ginzburg?Landau functional
6. Spectrum of the graph Laplacians
7. Gradient flow: Allen?Cahn
8. Merriman?Bence?Osher scheme
9. Graph curvature and mean curvature flow
10. Freezing of Allen?Cahn, MBO, and mean curvature flow
11. Multiclass extensions
12. Laplacian learning and Poisson learning
13. Conclusions
Bibliography.
Part of Cambridge Studies in Advanced Mathematics
available from April 2025
format: Hardback isbn: 9781009494113
Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.
Gives readers a complete introduction to oriented matroids from a unified perspective
Emphasizes the interplay of combinatorial, geometric, and topological ideas
Develops concepts using geometric intuition and assumes minimal prerequisites, including no background knowledge of matroid theory, to make the topic accessible to a broad audience
Allows researchers outside mathematics to approach the topic and its applications to data analysis
1. Realizable oriented matroids
2. Oriented matroids
3. Elementary operations and properties
4. The topological representation theorem
5. Strong maps and weak maps
6. Single-element extensions
7. The universality theorem
8. Oriented matroid polytopes
9. Subdivisions and triangulations
10. Spaces of oriented matroids
11. Hints on selected exercises
References
Index.
Part of Cambridge Series in Statistical and Probabilistic Mathematics
available from April 2025
format: Hardback isbn: 9781009521437
This extensive revision of the 2007 book 'Random Graph Dynamics,' covering the current state of mathematical research in the field, is ideal for researchers and graduate students. It considers a small number of types of graphs, primarily the configuration model and inhomogeneous random graphs. However, it investigates a wide variety of dynamics. The author describes results for the convergence to equilibrium for random walks on random graphs as well as topics that have emerged as mature research areas since the publication of the first edition, such as epidemics, the contact process, voter models, and coalescing random walk. Chapter 8 discusses a new challenging and largely uncharted direction: systems in which the graph and the states of their vertices coevolve.
Brings readers up to speed on recent developments
Concisely summarizes the state of the field in a succinct volume
Emphasizes probabilistic arguments
'This fully revised book showcases the enormous recent progress made in the field of random graphs and dynamics on them. The chosen topics, including small-world properties, random walks, and interacting particle systems, are carefully picked and well aligned. The author gives a high-level explanation of the proofs of their main results. Omitting the full proofs gives the reader insight in a wide range of topics and the tools of the trade for them, while keeping the book relatively short. Thus, it is an excellent starting point to this exciting field, with dozens of pointers to the literature for more details.' Remco van der Hofstad, Eindhoven University of Technology
'Random graph theory (preferential attachment graphs, small-world networks, configuration model, etc.) and interacting particle systems (contact process, voter model, etc.) are currently two of the most important branches of probability theory in terms of mathematics and their applications in epidemiology and sociology. Durrett's book Dynamics on Graphs covers both topics in parallel before merging them in an elegant way along with a rigorous mathematical treatment. The book concludes with the very hot topic of adaptive networks that combine dynamics on the graph and dynamics of the graph, opening the door to future challenging research. Great job!' Nicolas Lanchier, Arizona State University
Preface
Notation
1. Erd?s?Renyi random graphs
2. General degree distributions
3. Inhomogeneous random graphs
4. Epidemics
5. Contact process
6. Random walks, mixing times
7. Voter models, coalescing RWs
8. Coevolving systems
Appendix. Large deviations
Books and Long Surveys
Index.
available from April 2025
format: Paperback isbn: 9781009505888
This textbook focuses on general topology. Meant for graduate and senior undergraduate mathematics students, it introduces topology thoroughly from scratch and assumes minimal basic knowledge of real analysis and metric spaces. It begins with thought-provoking questions to encourage students to learn about topology and how it is related to, yet different from, geometry. Using concepts from real analysis and metric spaces, the definition of topology is introduced along with its motivation and importance. The text covers all the topics of topology, including homeomorphism, subspace topology, weak topology, product topology, quotient topology, coproduct topology, order topology, metric topology, and topological properties such as countability axioms, separation axioms, compactness, and connectedness. It also helps to understand the significance of various topological properties in classifying topological spaces.
Applications of connectedness and examples of compactness, connectedness, and path-connectedness provided in detail
Stepwise and detailed proofs for a better understanding of concepts
Figures and diagrams for ease of visualization
1. Preliminaries
2. Topological Spaces 3. Continuous Functions
4. Techniques of Creating Topologies: New from Old
5. The Topology of Metric Spaces
6. Countability Axioms
7. Separation Axioms
8. Compactness
9. Connectedness
Appendix: From General Topology to Algebraic Topology
References
Index.
Part of Cambridge Studies in Advanced Mathematics
available from May 2025
format: Hardback isbn: 9781009504553
Offering a comprehensive introduction to number theory, this is the ideal book both for those who want to learn the subject seriously and independently, or for those already working in number theory who want to deepen their expertise. Readers will be treated to a rich experience, developing the key theoretical ideas while explicitly solving arithmetic problems, with the historical background of analytic and algebraic number theory woven throughout. Topics include methods of solving binomial congruences, a clear account of the quantum factorization of integers, and methods of explicitly representing integers by quadratic forms over integers. In the later parts of the book, the author provides a thorough approach towards composition and genera of quadratic forms, as well as the essentials for detecting bounded gaps between prime numbers that occur infinitely often.
Contains non-trivial explicit examples throughout the volume, giving readers the confidence to solve numerically impressive problems
Precise historical information will teach readers the importance of knowing who did what, when, where, and how in academic research
A modestly theoretic approach throughout facilitates a self-contained volume suitable for independent study without the need for further reference sources
Preface
For readers
Table of theorems
1. Divisibility
2. Congruences
3. Characters
4. Quadratic forms
5. Distribution of prime numbers
Bibliography
Index.