Oxford Graduate Texts in Mathematics
the only book devoted to the topic of graph homomorphism
contains exercises of varying difficulty to aid learning
presents material from many different sources and develops the theory in an easily accessible form
new topics include graph convergence and limits, homomorphisms of strongly regular graphs, polymorphisms, and vector colourings
covers important new breakthroughs, such as a disproof of the Hedetniemi's Conjecture and the proof of the Feder-Vardi Conjecture
Graph theory and especially the study of graph homomorphisms has developed exponentially in the last couple of decades. Originally based on the authors' graduate course lecture notes, this is still the only book devoted entirely to graph homomorphisms, bringing together the highlights of the theory and its many applications.
It offers a useful perspective on more traditional graph theory topics such as graph reconstruction, various forms of graph colouring, graph products, automorphism groups, chordality, and interval graphs, and has applications in complexity theory, artificial intelligence, telecommunication, and statistical physics. It is intended to be a sampler of this rich subject and focuses on its most interesting results, techniques, and applications, including its algebraic, combinatorial, and algorithmic aspects.
The book discusses most of the recent developments in the field, such as the disproof of the Hedetniemi Product Conjecture, the proof of the Feder-Vardi Dichotomy Conjecture of constraint satisfaction problems, as well as the celebrated theory of graph convergence and graphons. The text contains exercises of varying difficulty, intended to support readers' learning and further development.
1:Introduction
2:Products, retracts, and polymorphisms
3:The partial order of graphs and homomorphisms
4: The structure of composition
5:Testing for the existence of homomorphisms
6:Colouring: variations on a theme
7:Recent highlights
From bestselling author and mathematician Ian Stewart, the fascinating story of the extreme problems that have driven math forward from antiquity to today
Many of the deepest and most important areas of mathematics have emerged from questions about extremes—the shortest path between two points on a curved surface, the smallest area spanning a wire, or the fewest colors needed to make a map. Mathematicians have been pushing restlessly toward extremes for thousands of years. The isoperimetric problem, for example—which asks for the shortest route enclosing a given area—can be traced to ancient Carthage. By contrast, it was only in 2017 that the densest ways to pack identical spheres into a 24-dimensional space was proven. In Reaching for the Extreme, bestselling author Ian Stewart, one of the world’s most popular writers on mathematics, presents a dazzling, wide-ranging tour of math’s outer limits.
Stewart tells the stories of sixteen superlative problems—their history, the struggles to solve them, and the uses of some of the results. From the biggest number to the smallest, the fastest fall to the weirdest symmetry, and the best fold to the shortest proof, these questions are either pure thought experiments or are motivated by real-world challenges. The Plateau problem, about the geometry of soap bubbles, led to the notion of a minimal surface—now used in cosmology, biology, and other fields. Meanwhile, the 2023 discovery of a single tile shape that covers the infinite plane without repeating the same pattern has no application—yet.
Reaching for the Extreme illuminates how mathematicians drive knowledge forward by reaching for the edges and solving some of the world’s most fascinating problems.
Format: Hardback, 465 pages, height x width: 235x155 mm, 4 Illustrations, color; 137 Illustrations, black and white
Pub. Date: 14-Dec-2025
ISBN-13: 9783032037329
This textbook is designed for a one-semester introductory course in Differential Geometry. It covers the fundamentals of differentiable manifolds, explores Lie groups and homogeneous spaces, and concludes with rigorous proofs of Stokes Theorem and the de Rham Theorem. The material closely follows the author's lectures at ETH Zürich
Chapter 1. Smooth manifolds.
Chapter 2. Tangent spaces.
Chapter 3. Partition of unity.
Chapter 4. The derivative.
Chapter 5. The tangent bundle.
Chapter 6. Submanifolds.
Chapter 7. The Whitney theorems.
Chapter 8. Vector fields.
Chapter 9. Flows.
Chapter 10. Lie groups.
Chapter 11. The Lie algebra of a Lie group.
Chapter 12. Smooth actions of Lie groups.-
Chapter 13. Homogeneous spaces.
Chapter 14. Distributions and integrability.
Chapter 15. Foliations and the Frobenius theorem.
Chapter 16. Bundles.
Chapter 17. The fibre bundle construction theorem.
Chapter 18. Associated bundles.
Chapter 19. Tensor and exterior algebras.
Chapter 20. Sections of vector bundles.
Chapter 21. Tensor fields.
Chapter 22. The Lie derivative revisited.
Chapter 23. The exterior differential.
Chapter 24. Orientations and manifolds with boundary.
Chapter 25. Smooth singular cubes.
Chapter 26. Stokes' theorem.
Chapter 27. The Poincaré lemma and the de Rham theorem.
Format: Hardback, 200 pages, height x width: 235x155 mm, 28 Illustrations, color; 6 Illustrations, black and white
Series: Interdisciplinary Applied Mathematics
Pub. Date: 25-Nov-2025
ISBN-13: 9783032048257
This text provides a modern introduction to the mathematical formulation and physical applications of Stefan problems. With a careful balance of theory and practice, it is suitable for both graduate students and experienced researchers in applied math, engineering, physics, and chemistry. The formulation of the Stefan problem and several analytical and approximate solution methods are described in the first three chapters. Applied mathematical techniques needed for later chapters, such as non-dimensionalization, perturbation methods, and lubrication theory, are also covered. The remaining chapters are more specialized and explore formulations going beyond the classical Stefan problem, for example where the material properties and phase change temperatures vary. The theory is always motivated by physical situations and examples: phase change with a flowing liquid in the context of microvalves and ice accretion on aircraft; the solidification of a supercooled liquid, the melting or growth of nanoparticles and nanocrystals and phase change when the heat flow no longer follows Fourier’s law.
Introduction.- Exact and approximate solutions.- Solidification of a
thin liquid layer.- Variable property Stefan problem.- Variable interface
conditions.- Non-Fourier Stefan problems.- Hints to Exercises.- Index.
Format: Hardback, 376 pages, height x width: 235x155 mm, 134 Illustrations, black and white
Series: Texts in Applied Mathematics
Pub. Date: 17-Dec-2025
ISBN-13: 9783032040824
This textbook offers a rigorous yet accessible introduction to the qualitative theory of dynamical systems, focusing on both discrete- and continuous-time systemsthose defined by iterated maps and differential equations. With clarity and precision, it provides a conceptual framework and the essential tools needed to describe, analyze, and understand the behavior of real-world systems across the sciences and engineering.
Designed for advanced undergraduates and early graduate students, the book assumes only a foundational background in analysis, linear algebra, and differential equations. It bridges the gap between introductory courses and more advanced treatments by offering a self-contained and balanced approachone that integrates geometric intuition with analytical rigor.
A carefully curated selection of topics essential for applied contexts Full, detailed proofs of cornerstone results, including the Poincaré-Bendixson theorem, Lyapunovs stability criteria, Grobman-Hartman theorem, Center Manifold theorem A unified treatment of discrete- and continuous-time systems, with discrete methods often paving the way for their continuous counterparts Employing modern functional analytic techniques to streamline and clarify complex arguments Special attention to invariant manifolds, symbolic dynamics, and topological normal forms for codimension-one bifurcations
Whether for students planning further study in pure or applied mathematics,
or for those in disciplines such as physics, biology, or engineering seeking
to apply dynamical systems theory in practice, this book offers a concise
yet comprehensive entry point. Instructors will appreciate its modular
structure and completeness, while students will benefit from its clarity,
rigor, and insightful presentation
Introduction.- Linear maps and ODEs.- Local behavior of nonlinear
systems.- Planar ODE.- etc.
Format: Hardback, 308 pages, height x width: 235x155 mm, XIX, 308 p.
Series: Texts and Readings in Mathematics
Pub. Date: 09-Nov-2025
ISBN-13: 9789819527571
This book can serve as a first course on measure theory and measure theoretic probability for upper undergraduate and graduate students of mathematics, statistics and probability. Starting from the basics, the measure theory part covers Caratheodorys theorem, LebesgueStieltjes measures, integration theory, Fatous lemma, dominated convergence theorem, basics of Lp spaces, transition and product measures, Fubinis theorem, construction of the Lebesgue measure in Rd, convergence of finite measures, JordanHahn decomposition of signed measures, RadonNikodym theorem and the fundamental theorem of calculus.
The material on probability covers standard topics such as BorelCantelli lemmas, behaviour of sums of independent random variables, 0-1 laws, weak convergence of probability distributions, in particular via moments and cumulants, and the central limit theorem (via characteristic function, and also via cumulants), and ends with conditional expectation as a natural application of the RadonNikodym theorem. A unique feature is the discussion of the relation between moments and cumulants, leading to Isserlis formula for moments of products of Gaussian variables and a proof of the central limit theorem avoiding the use of characteristic functions.
For clarity, the material is divided into 23 (mostly) short chapters. At
the appearance of any new concept, adequate exercises are provided to strengthen
it. Additional exercises are provided at the end of almost every chapter.
A few results have been stated due to their importance, but their proofs
do not belong to a first course. A reasonable familiarity with real analysis
is needed, especially for the measure theory part. Having a background
in basic probability would be helpful, but we do not assume a prior exposure
to probability
Preliminaries.- Classes of Sets.- Introduction to Measures.- Extension
of Measures.- Lebesgue-Stieltjes Measures.- Measurable
Functions.- Integral.- Basic Inequalities.- Lp Spaces: Topological
Properties.- Product Spaces and Transition Measures.- Random Variables and
Vectors.- Moments and Cumulants.- Further Modes of Convergence of
Functions.- Independence and Basic Conditional Probability.- 0-1 Laws.- Sums
of Independent Random Variables.- Convergence of Finite
Measures.- Characteristic Functions.- Central Limit Theorem.- Signed
Measure.- Randon-Nikodym Theorem .- Fundamental Theorem of Calculus.-
Conditional Expectation.
Format: Hardback, 550 pages, height x width: 235x155 mm, 2 Illustrations, black and white
Series: Grundlehren der mathematischen Wissenschaften
Pub. Date: 14-Nov-2025
ISBN-13: 9783032040985
This monograph studies duality in interacting particle systems, a topic combining probability theory, statistical physics, Lie algebras, and orthogonal polynomials. It offers the first comprehensive account of duality theory in the context of interacting particle systems.
Using a Lie algebraic framework, the book demonstrates how dualities arise in families of systems linked to algebraic representations. The exposition centers on three key processes: independent random walks, the inclusion process, and the exclusion processassociated with the Heisenberg, su(1,1), and su(2) algebras, respectively. From these three basic cases, several new processes and their duality relations are derived. Additional models, such as the Brownian energy process, the KMP model and the Kac model, are also discussed, along with topics like the hydrodynamic limit and non-equilibrium behavior. Further, integrable systems associated to the su(1,1) algebra are studied and their non-equilibrium steady states are computed.
Intentionally accessible and self-contained, this book is aimed at graduate-level researchers and also serves as a comprehensive introduction to the duality of Markov processes and beyond.
Chapter 1. Introduction.
Chapter 2. Basics of the algebraic approach.-
Chapter 3. Duality for independent random walkers: part 1.
Chapter 4. Duality for independent random walkers: part 2.
Chapter 5. Duality for the symmetric inclusion process.
Chapter 6. Duality for the Brownian energy process.
Chapter 7. Duality for the symmetric partial exclusion process.-
Chapter 8. Duality for other models.
Chapter 9. Orthogonal dualities.-
Chapter 10. Consistency.
Chapter 11. Duality for non-equilibrium systems.-
Chapter 12. Duality and macroscopic fields.
Chapter 13. Duality and
integrable models.