The culmination of years of teaching experience, this book provides a modern introduction to the mathematical theory of interacting particle systems. Assuming a background in probability and measure theory, it has been designed to support a one-semester course at a Master or Ph.D. level. It also provides a useful reference for researchers, containing several results that have not appeared in print in this form before. An emphasis is placed on graphical representations, which are used to give a construction that is intuitively easier to grasp than the traditional generator approach. Also included is an extensive look at duality theory, along with discussions of mean-field methods, phase transitions and critical behaviour. The text is illustrated with the results of numerical simulations and features exercises in every chapter. The theory is demonstrated on a range of models, reflecting the modern state of the subject and highlighting the scope of possible applications.

Preface; 1. Introduction; 2. Continuous-time Markov chains; 3. The mean-field limit; 4. Construction and ergodicity; 5. Monotonicity; 6. Duality; 7. Oriented percolation; Bibliography; Index.

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.

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.

'Acta Numerica' is an annual publication containing invited survey papers by leading researchers in numerical mathematics and scientific computing. The papers present overviews of recent developments in their area and provide state-of-the-art techniques and analysis.

1. Cut finite element methods Erik Burman, Peter Hansbo, Mats G. Larson and Sara Zahedi; 2. Ensemble Kalman methods: a mean-field perspective Edoardo Calvello, Sebastian Reich and Andrew M. Stuart; 3. The discontinuous Petrov–Galerkin method Leszek Demkowicz and Jay Gopalakrishnan; 4. Time parallelization for hyperbolic and parabolic problems Martin J. Gander, Shu-Lin Wu and Tao Zhou; 5. Optimization problems governed by systems of PDEs with uncertainties Matthias Heinkenschloss and Drew P. Kouri; 6. Distributionally robust optimization Daniel Kuhn, Soroosh Shafiee and Wolfram Wiesemann; 7. Acceleration methods for fixed-point iterations Yousef Saad; 8. Sparse linear least-squares problems Jennifer Scott and Miroslav Tůma.

Isabella Verdinelli, Larry Wasserman

This comprehensive modern look at regression covers a wide range of topics and relevant contemporary applications, going well beyond the topics covered in most introductory books. With concision and clarity, the authors present linear regression, nonparametric regression, classification, logistic and Poisson regression, high-dimensional regression, quantile regression, conformal prediction and causal inference. There are also brief introductions to neural nets, deep learning, random effects, survival analysis, graphical models and time series. Suitable for advanced undergraduate and beginning graduate students, the book will also serve as a useful reference for researchers and practitioners in data science, machine learning, and artificial intelligence who want to understand modern methods for data analysis.

Preface; Notation; 1. Introduction; 2. Linear regression; 3. Prediction error, cross-validation and model selection; 4. High dimensional linear regression; 5. Logistic and Poisson regression; 6. Univariate nonparametric regression; 7. Nonparametric regression with multiple features; 8. Quantile regression; 9. Classification; 10. Prediction sets and conformal inference; 11. Causal inference; 12. Other topics; Appendix A. Matrix theory; Appendix B. Basic probability and statistics; Data Sources; References; Index.

Francesco Viola, Marco D. de Tullio, Roberto Verzicco

Unlock the potential of computational fluid dynamics with this essential guide for master's and graduate students, and researchers. It explores the immersed boundary method (IBM), a revolutionary approach for simulating flows in complex geometries. With a focus on fluid/structure interaction, it examines theoretical principles and practical implementations, offering insights into tackling intricate geometries and enhancing simulation accuracy. The book features a series of numerical examples that increase in complexity, and is accompanied by the source code, allowing readers to replicate results and deepen their understanding. Whether you're wanting to refine your skills or embark on new research, this introduction will empower you to master the art of complex flow simulations.

1. Introduction; 2. Introduction to flow simulations; 3. Computational geometry and grid generation; 4. Forcing methods; 5. Moving boundaries; 6. Computation of hydrodynamic loads; 7. Fluid–structure interaction; 8. Turbulence and wall models within IBMs; 9. Advanced IBM applications; 10. Numerical examples; References; Index.

Over the past few decades, graph theory has developed into one of the central areas of modern mathematics, with close (and growing) connections to areas of pure mathematics such as number theory, probability theory, algebra and geometry, as well as to applied areas such as the theory of networks, machine learning, statistical physics, and biology. It is a young and vibrant area, with several major breakthroughs having occurred in just the past few years. This book offers the reader a gentle introduction to the fundamental concepts and techniques of graph theory, covering classical topics such as matchings, colourings and connectivity, alongside the modern and vibrant areas of extremal graph theory, Ramsey theory, and random graphs. The focus throughout is on beautiful questions, ideas and proofs, and on illustrating simple but powerful techniques, such as the probabilistic method, that should be part of every young mathematician's toolkit.

Introduction; 1. Basic graph theory; 2. Extremal graph theory; 3. Classical graph theory; 4. Ramsey theory; 5. Random graphs.

Over the past few decades, graph theory has developed into one of the central areas of modern mathematics, with close (and growing) connections to areas of pure mathematics such as number theory, probability theory, algebra and geometry, as well as to applied areas such as the theory of networks, machine learning, statistical physics, and biology. It is a young and vibrant area, with several major breakthroughs having occurred in just the past few years. This book offers the reader a gentle introduction to the fundamental concepts and techniques of graph theory, covering classical topics such as matchings, colourings and connectivity, alongside the modern and vibrant areas of extremal graph theory, Ramsey theory, and random graphs. The focus throughout is on beautiful questions, ideas and proofs, and on illustrating simple but powerful techniques, such as the probabilistic method, that should be part of every young mathematician's toolkit.

Introduction; 1. Basic graph theory; 2. Extremal graph theory; 3. Classical graph theory; 4. Ramsey theory; 5. Random graphs.

This book provides the first comprehensive study of the geometric aspects of Manin's conjecture. It equips the reader with a working knowledge of higher dimensional algebraic geometry, including the minimal model program and its applications to arithmetic and Diophantine geometry. The text also develops the foundations of the moduli theory of rational curves on Fano varieties and explores its role in the geometric formulation of Manin's conjecture, supported by worked examples. The book is suitable for graduates and researchers in arithmetic geometry seeking a modern introduction to birational geometry and the moduli theory of rational curves. It will also interest experts in higher‑dimensional algebraic geometry who wish to understand recent applications of these techniques to arithmetic geometry.

Preface; Acknowledgements; 1. Introduction; 2. Height functions; 3. Quick introduction to higher-dimensional algebraic geometry; 4. Manin's conjecture; 5. Proofs of main results; 6. Further examples; 7. Moduli of rational curves; 8. Examples of geometric Manin's conjecture II; References; Index.

Abductive reasoning is a form of inference that infers some hypothesis because of what that hypothesis explains. Unlike deductive reasoning, it yields a plausible conclusion but does not definitively verify it. The theory of compositional abduction developed in this book provides a novel theory of confirmation. Aizawa uses case studies to analyse how scientists interpret the results of experiments to support compositional hypotheses (hypotheses about what things are composed of) and suggests that they use a kind of abduction. His theory is offered as an alternative account of scientific reasoning that the logical empiricists would have interpreted as hypothetico-deductive confirmation. It is also an alternative to the Peircean interpretation of the role of abduction in science. It will be valuable to philosophers of science, those working on hypothetico-deductive confirmation, Peirce's view of abduction, inference to the best explanation, and the New Mechanism. This title is also available as Open Access on Cambridge Core.

1. The metatheory of the project; 2. Singular compositional explanations; 3. Singular compositional explanations: special cases; 4. Singular compositional abduction; 5. The compositional basis of resting and action potentials; 6. The hermann grid illusion; 7. Peirce and the neo-peirceans; 8. Harman and inference to the best explanation; 9. Lipton and inference to the best explanation; 10. Manipulationist new mechanism; 11. Conclusion.

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.

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.

A valuable resource for researchers in discrete and combinatorial geometry, this book offers comprehensive coverage of several modern developments on algebraic and combinatorial properties of polytopes. The introductory chapters provide a new approach to the basic properties of convex polyhedra and how they are connected; for instance, fibre operations are treated early on. Finite tilings and polyhedral convex functions play an important role, and lead to the new technique of tiling diagrams. Special classes of polytopes such as zonotopes also have corresponding diagrams. A central result is the complete characterization of the possible face-numbers of simple polytopes. Tools used for this are representations and the weight algebra of mixed volumes. An unexpected consequence of the proof is an algebraic treatment of Brunn–Minkowski theory as applied to polytopes. Valuations also provide a thread running through the book, and the abstract theory and related tensor algebras are treated in detail.

Algebra; 1. Polyhedra; 2. Linear systems; 3. Representations; 4. Polyhedral functions; 5. Finite tilings; 6. Polytopes; 7. Refinements in polytopes; 8. Numbers of faces; 9. Polytopes with symmetry; 10. Zonotopes; 11. Infinite tilings; 12. Volume and its relatives; 13. Scalar weight algebra; 14. Simple polytopes; 15. Brunn–Minkowski theory; 16. Algebra of polyhedra; 17. Polytope ring; 18. Polytope algebra; 19. Tensor weights; 20. Fibre algebra; 21. Lattice polytopes and valuations; Afterword; References; Notation index; Author index; Subject index.

This rich, deeply researched study offers the first comprehensive exploration of cross-cultural plant knowledge in eighteenth-century Mauritius. Using the concept of creolisation – the process by which elements of different cultures are brought together to create entangled and evolving new entities – Brixius examines the production of knowledge on an island without long-established traditions of botany as understood by Europeans. Once foreign plants and knowledge arrived in Mauritius, they were adapted to new environmental circumstances and a new socio-cultural space. Brixius explores how French colonists, settlers, mediators, labourers and enslaved people experienced and shaped the island's botanical past, centring the contributions of subaltern actors. By foregrounding neglected non-European actors from both Africa and Asia, within a melting pot of cultivation traditions from around the world, she presents a truly global history of botanical knowledge.

Introduction; 1. The limits of French colonial visions and science; 2. The acquisition of knowledge and plants, from Madagascar to China; 3. Agriculture and everyday knowledge; 4. Enslaved people as knowledge carriers; 5. The cross-cultural quest for spices in Southeast Asia; 6. Materials, environment, and the application of knowledge; Conclusion.

Charles Darwin is known as a biologist, geologist, and naturalist, but he was also a philosopher. This book uncovers Darwin's forgotten philosophical theory of emotion, which combines earlier associationist theories with his theory of evolution. The British associationists of the eighteenth and nineteenth centuries argued that the mind operates primarily through the association of ideas, and that emotions are strings of thoughts, feelings, and outward expressions, connected by habit and association. Charles Darwin's early notebooks on emotion reveal a keen interest in associationist philosophy. This book shows that one of his final works, The Expression of the Emotions in Man and Animals (1872), is a work of associationist philosophy, and analyzes Darwin's revolutionary idea: that if the associations that produce emotions can be inherited, then the theory of evolution can explain how emotions first occurred in simpler organisms and then developed and were compounded into the complex experiences humans have today.

Introduction; 1. Darwin's expression; 2. Darwin's predecessors; 3. Darwin's associationism; 4. Darwin's emotions; 5. Darwin's expressions; 6. Darwin's lost legacy; References.

Exactly a decade after the publication of the Sz.-Nagy dilation theorem, Tsuyoshi Ando 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 Ando'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 bivariate setting. Sixty years since the appearance of Ando's first step comes this thorough systematic treatment of a dilation and model theory 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. Ando's dilation and commutant lifting theorems; 5. Douglas-type model for Ando isometric lifts; 6. Schäffer-type model for Ando isometric lifts; 7. Strongly minimal Ando 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.

Dynamics of Mechanical Systems: A Modern Introduction to Three-Dimensional Rigid Body Dynamics

Written by experts in the field, this text provides a modern introduction to three-dimensional dynamics for multibody systems. It covers rotation matrices, the twist-wrench formalism for multibody dynamics and Lagrangian dynamics, an approach that is often overlooked at the undergraduate level. The only prerequisites are differential equations and linear algebra as covered in a first-year engineering mathematics course. The text focuses on obtaining and understanding the equations of motion, featuring a rich set of examples and exercises that are drawn from real-world scenarios. Readers develop a reliable physical intuition that can then be used to apply dynamic analysis software tools, and to develop simplified approximate models. With this foundation, they will be able to confidently use the equations of motion in a variety of applications, ranging from simulation and design to motion planning and control.

1. Introduction; 2. Kinematics; 3. Particle dynamics; 4. Dynamics of a rigid body; 5. Multibody dynamics: a twist–wrench formulation; 6. Lagrangian dynamics; 7. Linearized dynamics; Appendix. Linear algebra and matrix theory basics; Index.

B. Surendranath Reddy, M. Evangeline Prathibha, T. Asir

Designed for undergraduate students of computer science, mathematics, and engineering, this book provides the tools and understanding needed to master graph theory and algorithms. It offers a strong theoretical foundation, detailed pseudocodes, and a range of real-world and illustrative examples to bridge the gap between abstract concepts and practical applications. Clear explanations and chapter-wise exercises support ease of comprehension for learners. The text begins with the basic properties of graphs and progresses to topics such as trees, connectivity, and distances in graphs. It also covers Eulerian and Hamiltonian graphs, matchings, planar graphs, and graph colouring. The book concludes with discussions on independent sets, the Ramsey theorem, directed graphs and networks. Concepts are introduced in a structured manner, with appropriate context and support from mathematical language and diagrams. Algorithms are explained through rules, reasoning, pseudocode, and relevant examples.

Preface; 1. Introduction to graphs; 2. Basic properties of graphs; 3. Trees; 4. Connectivity; 5. Distance in Graphs; 6. Eulerian graphs and Hamiltonian graphs; 7. Matchings; 8. Planar graphs; 9. Coloring of Graphs; 10. Independent sets and Ramsey theory; 11. Directed Graphs; Bibliography; Index.

Dhruv Mubayi, Jacques A. Verstraete

Providing a cohesive reference for advanced undergraduates, graduate students and even experienced researchers, this text contains both introductory and advanced material in extremal graph theory, hypergraph theory and Ramsey theory. Along the way, the book includes many modern proof techniques in the field such as the probabilistic method and algebraic methods. Several recent breakthroughs are presented with complete proofs, for example, recent results on the sunflower problem, and off-diagonal and geometric Ramsey theory. It is perhaps unique in containing material on both hypergraph regularity and containers. Featuring an extensive list of exercises, the text is suitable as a teaching text for a variety of courses in extremal combinatorics. Each of the two parts can form the basis of separate courses, and the majority of sections are designed to match the length of a single lecture.

Part I. Extremal Graph Theory: Introduction to Part I; 1. Matchings, paths and cycles; 2. Probabilistic methods and random graphs; 3. Turán's theorem; 4. Extremal problems for bipartite graphs; 5. Moore graphs and the even cycle theorem; 6. Random constructions; 7. Szemerédi's regularity lemma; 8. Pseudorandom graphs; 9. Graph Ramsey theory; 10. pseudorandom graph constructions in Ramsey theory; Part II. Extremal Set Theory and Hypergraph Theory: Introduction to Part II; 11. Posets; 12. Intersecting families; 13. Shadows and matchings; 14. Linear algebra method; 15. Algebraic methods; 16. Sunflowers; 17. The Turán problem for hypergraphs; 18. Matchings in linear hypergraphs; 19. Independent sets in hypergraphs; 20. Hypergraph regularity; 21. Hypergraph containers; 22. Hypergraph Ramsey theory; References; Glossary of notation; Index.

Ellen Powell, Nathanaël Berestycki

In this comprehensive volume, the authors introduce some of the most important recent developments at the intersection of probability theory and mathematical physics, including the Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity. This is the first book to present these topics using a unified approach and language, drawing on a large array of multi-disciplinary techniques. These range from the combinatorial (discrete Gaussian free field, random planar maps) to the geometric (culminating in the path integral formulation of Liouville conformal field theory on the Riemann sphere) via the complex analytic (based on the couplings between Schramm–Loewner evolution and the Gaussian free field). The arguments (currently scattered over a vast literature) have been streamlined and the exposition very carefully thought out to present the theory as much as possible in a reader-friendly, pedagogical yet rigorous way, suitable for graduate students as well as researchers.

1. Definition and properties of the GFF; 2. Liouville measure; 3. Gaussian multiplicative chaos; 4. Statistical physics on random planar maps; 5. Introduction to Liouville conformal field theory; 6. Gaussian free field with Neumann boundary conditions; 7. Quantum wedges and scale-invariant random surfaces; 8. SLE and the quantum zipper; 9. Liouville quantum gravity as a mating of trees; A. Chordal Loewner chains and chordal SLE; B. Reverse Loewner flow and reverse SLE; C. Radial Loewner chains and radial SLE; D. Convergence of random variables in the space of distributions; References; Notation and symbols; Subject index.

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.

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.

'High-Dimensional Probability,' winner of the 2019 PROSE Award in Mathematics, offers an accessible and friendly introduction to key probabilistic methods for mathematical data scientists. Streamlined and updated, this second edition integrates theory, core tools, and modern applications. Concentration inequalities are central, including classical results like Hoeffding's and Chernoff's inequalities, and modern ones like the matrix Bernstein inequality. The book also develops methods based on stochastic processes – Slepian's, Sudakov's, and Dudley's inequalities, generic chaining, and VC-based bounds. Applications include covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, and machine learning. New to this edition are 200 additional exercises, alongside extra hints to assist with self-study. Material on analysis, probability, and linear algebra has been reworked and expanded to help bridge the gap from a typical undergraduate background to a second course in probability.

Foreword Sara van de Geer; Preface; Appetizer. Using probability to cover a set; 1. A quick refresher on analysis and probability; 2. Concentration of sums of independent random variables; 3. Random vectors in high dimensions; 4. Random matrices; 5. Concentration without independence; 6. Quadratic forms, symmetrization, and contraction; 7. Random processes; 8. Chaining; 9. Deviations of random matrices on sets; Hints for the exercises; References; Index.

Statistical modelling and machine learning offer a vast toolbox of inference methods with which to model the world, discover patterns and reach beyond the data to make predictions when the truth is not certain. This concise book provides a clear introduction to those tools and to the core ideas – probabilistic model, likelihood, prior, posterior, overfitting, underfitting, cross-validation – that unify them. Toy and real examples illustrate diverse applications ranging from biomedical data to treasure hunts, while the accompanying datasets and computational notebooks in R and Python encourage hands-on learning. Instructors can benefit from online lecture slides and solutions to all the exercises. Requiring only first-year university-level knowledge of calculus, probability and linear algebra, the book equips students in statistics, data science and machine learning, as well as those in quantitative applied and social science programmes, with the tools and conceptual foundations to explore more advanced techniques.

1. Orientation; 2. Supervised learning warm-up; 3. Unsupervised learning warm-up; 4. Interlude: probability, likelihood and Bayes; 5. Probabilistic modelling; 6. Frequentist and Bayesian uncertainty; 7. Frequentist linear regression; 8. Directed graphical models; 9. Bayesian linear regression, priors, and regularisation; 10. Bayesian methods; 11. Classification; 12. Unsupervised learning: a deeper dive; 13. Neural networks and deep learning; 14. Expanding the toolkit; A. Probability theory; B. Linear algebra; C. Jensen's and Gibbs' inequalities; References; Index.

Statistical modelling and machine learning offer a vast toolbox of inference methods with which to model the world, discover patterns and reach beyond the data to make predictions when the truth is not certain. This concise book provides a clear introduction to those tools and to the core ideas – probabilistic model, likelihood, prior, posterior, overfitting, underfitting, cross-validation – that unify them. Toy and real examples illustrate diverse applications ranging from biomedical data to treasure hunts, while the accompanying datasets and computational notebooks in R and Python encourage hands-on learning. Instructors can benefit from online lecture slides and solutions to all the exercises. Requiring only first-year university-level knowledge of calculus, probability and linear algebra, the book equips students in statistics, data science and machine learning, as well as those in quantitative applied and social science programmes, with the tools and conceptual foundations to explore more advanced techniques.

1. Orientation; 2. Supervised learning warm-up; 3. Unsupervised learning warm-up; 4. Interlude: probability, likelihood and Bayes; 5. Probabilistic modelling; 6. Frequentist and Bayesian uncertainty; 7. Frequentist linear regression; 8. Directed graphical models; 9. Bayesian linear regression, priors, and regularisation; 10. Bayesian methods; 11. Classification; 12. Unsupervised learning: a deeper dive; 13. Neural networks and deep learning; 14. Expanding the toolkit; A. Probability theory; B. Linear algebra; C. Jensen's and Gibbs' inequalities; References; Index.

This up-to-date introduction to type theory and homotopy type theory will be essential reading for advanced undergraduate and graduate students interested in the foundations and formalization of mathematics. The book begins with a thorough and self-contained introduction to dependent type theory. No prior knowledge of type theory is required. The second part gradually introduces the key concepts of homotopy type theory: equivalences, the fundamental theorem of identity types, truncation levels, and the univalence axiom. This prepares the reader to study a variety of subjects from a univalent point of view, including sets, groups, combinatorics, and well-founded trees. The final part introduces the idea of higher inductive type by discussing the circle and its universal cover. Each part is structured into bite-size chapters, each the length of a lecture, and over 200 exercises provide ample practice material.

Preface; Introduction; Part I. Martin-Löf's Dependent Type Theory: 1. Dependent type theory; 2. Dependent function types; 3. The natural numbers; 4. More inductive types; 5. Identity types; 6. Universes; 7. Modular arithmetic via the Curry-Howard interpretation; 8. Decidability in elementary number theory; Part II. The Univalent Foundations of Mathematics: 9. Equivalences; 10. Contractible types and contractible maps; 11. The fundamental theorem of identity types; 12. Propositions, sets, and the higher truncation levels; 13. Function extensionality; 14. Propositional truncations; 15. Image factorizations; 16. Finite types; 17. The univalence axiom; 18. Set quotients; 19. Groups in univalent mathematics; 20. General inductive types; Part III. The Circle: 21. The circle; 22. The universal cover of the circle; References; Index.

This book provides a clear and accessible introduction to ring theory for undergraduate students. Aligned with standard curricula, it simplifies abstract concepts through structured explanations, practical examples, and real-world applications. Ideal for both students and instructors, it serves as a valuable resource for mastering fundamental concepts in ring theory with ease. The text begins with an introduction to rings and goes on to cover subrings, integral domains, ideals, and factor rings. It also discusses ring homomorphisms and polynomial rings. The book concludes with topics such as polynomial factorization and divisibility in integral domains. Each chapter is supplemented with solved examples to foster a deeper understanding of the subject. A set of practice questions is also provided to sharpen problem-solving skills.

Preface; Preliminaries; Chapter 1. Introduction to Rings; Chapter 2. Subrings; Chapter 3. Integral Domains; Chapter 4. Ideals; Chapter 5. Ring Homomorphisms; Chapter 6. Polynomial Rings; Chapter 7. Factorization of Polynomials Chapter 8. Divisibility in Integral Domains; Index.

The study of smooth embeddings of 3-manifolds in 4-space has been hampered by difficulties with the simplest case, that of homology spheres. This book presents some advantages of working with locally flat embeddings. The first two chapters outline the tools used and give general results on embeddings of 3-manifolds in S4. The next two chapters consider which Seifert manifolds may embed, with criteria in terms of Seifert data. After summarizing results on those Seifert manifolds that embed smoothly, the following chapters determine which 3-manifolds with virtually solvable fundamental groups embed. The final three chapters study the complementary regions. When these have 'good' fundamental groups, topological surgery may be used to find homeomorphisms. Figures throughout help illustrate links representing embeddings and open questions are further discussed in the appendices, making this a valuable resource for graduate students and research workers in geometric topology.

1. Preliminaries; 2. Invariants and constructions; 3. 3-manifolds with S1-actions; 4. Seifert manifolds with non-orientable base orbifolds; 5. Smooth embeddings; 6. 3-manifolds with restrained fundamental group; 7. The complementary regions; 8. Abelian embeddings; 9. Nilpotent embeddings; Appendix A. The linking pairings of orientable Seifert manifolds; Appendix B. Homologically balanced nilpotent groups; Appendix C. Some questions; References; Index.

Cubature rules are indispensable tools in scientific computing and applied sciences whenever evaluating or discretizing integrals is needed. This monograph is the first comprehensive resource devoted to cubature rules in English since Stroud's classic 1971 book, and the first book about minimal cubature rules. The book explores the subject's theoretical side, which intersects with many branches of mathematics. Minimal cubature rules are intimately connected with common zeros of orthogonal polynomials, which can be described via the polynomial ideals and varieties. Many prominent or practical cubature rules are invariant under a finite group, and some involve symmetric functions and the discrete Fourier transform. Based on state-of-the-art research, the book systematically studies Gauss and minimal cubature rules, and includes a chapter on the practical aspects of construction cubature rules on triangles and simplexes. This comprehensive guide is ideal for researchers and advanced graduate students across the computational and applied mathematics community.

Preface; 1. Quadrature rules and orthogonal polynomials; 2. Cubature rules: basics; 3. Orthogonal polynomials of several variables; 4. Gauss cubature rules; 5. Lower bounds for the number of nodes; 6. First minimal cubature rules; 7. Further minimal cubature rules; 8. Discrete Fourier transform and cubature rules; 9. Cubature rules and polynomial ideals; 10. Epilogue: two open problems; 11. Addendum: cubature rules on triangle and simplex; References; Index.

Rajen D. Shah, Richard J. Samworth

Modern Statistical Methods and Theory: An Introduction to Nonparametric and High-Dimensional Statistics

Based on courses taught at the University of Cambridge, this text presents core contemporary statistical methods and theory in an accessible, self-contained and rigorous fashion, with a focus on finite-sample guarantees as opposed to asymptotic arguments. Many of the topics and results have not appeared in book form previously, and some constitute new research. The prerequisites are relatively light (primarily a good grasp of linear algebra and real analysis) and complete solutions to all 250+ exercises are available online. It is the perfect entry point to the subject for master's and graduate-level students in statistics, data science and machine learning, as well as related disciplines such as artificial intelligence, signal processing, information theory, electrical engineering and econometrics. Researchers in these fields will also find it an invaluable resource. This title is also available as Open Access on Cambridge Core.

Acknowledgements; 1. Introduction; 2. Linear models and ordinary least squares; 3. High-dimensional linear regression; 4. Kernel density estimation; 5. Nonparametric regression; 6. Reproducing kernel Hilbert spaces and kernel machines; 7. Conditional independence, graphical models and causality; 8. Minimax lower bounds; 9. Shape-constrained estimation; Appendix. Mathematical background; References; Index.

Alexander H. W. Schmitt, Frank Neumann, Pedro L. del Ángel R.

The present volume features contributions from the 2022 BIRS-CMO workshop 'Moduli, Motives and Bundles – New Trends in Algebraic Geometry' held at the Casa Matemática Oaxaca (CMO), in partnership with the Banff International Research Station for Mathematical Innovation and Discovery (BIRS). The first part presents overview articles on enumerative geometry, moduli stacks of coherent sheaves, and torsors in complex geometry, inspired by related mini course lecture series of the workshop. The second part features invited contributions by experts on a diverse range of recent developments in algebraic geometry, and its interactions with number theory and mathematical physics, offering fresh insights into this active area. Students and young researchers will appreciate this text's accessible approach, as well as its focus on future research directions and open problems.

Preface; 1. On the stack of 0-dimensional coherent sheaves: structural aspects Barbara Fantechi and Andrea T. Ricolfi; 2. Three lectures on quadratic enumerative geometry Marc Levine; 3. Flat torsors in complex geometry Juan Sebastián Numpaque-Roa and Florent Schaffhauser; 4. Universal chern classes on the moduli of bundles Donu Arapura; 5. Linear stability of coherent systems and applications to Butler's conjecture Abel Castorena, George H. Hitching, and Erick Luna; 6. A Kobayashi-Hitchin correspondence for general coherent systems Cesare Goretti; 7. On the injectivity and non-injectivity of the l-adic cycle class maps Bruno Kahn; 8. A primer on zeta functions and decomposition spaces Andrew Kobin; 9. Atiyah-Bott localization in equivariant Witt cohomology Marc Levine; 10. Base loci, semiampleness, and parallelizable manifolds Ernesto C. Mistretta; 11. Recent developments on compactifications of stacks of shtukas Tuan Ngo Dac and Yakov Varshavsky; 12. A common generalisation of the André-Oort and André-Pink-Zannier conjectures Rodolphe Richard and Andrei Yafaev.

A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. This book answers this question positively by constructing an enriched multifunctor from the G-categorically enriched multicategory of O-pseudoalgebras to the symmetric monoidal category of orthogonal G-spectra, for a compact Lie group G and a 1-connected pseudo-commutative G-categorical operad O. As the main application of its enriched multifunctoriality, this machine preserves all equivariant algebraic structures parametrized by multicategories enriched in either G-spaces or G-categories. In addition to highly detailed proofs of all the main results, this work also reviews relevant concepts, including orthogonal G-spectra. Appendices provide the necessary background on symmetric monoidal categories, multicategories, and classifying space. Clearly written and well organized, this book will be a useful reference for researchers and graduate students for years to come.

Part I. From Operadic Pseudoalgebras to G-G-Categories: 1. Pseudo-commutative operads and operadic pseudoalgebras; 2. GCat-multicategories of operadic pseudoalgebras; 3. Symmetric monoidal closed category of g-g-categories; 4. GCat-multifunctors from operadic pseudoalgebras to g-g-categories; Part II. GCat-Multifunctors from Operadic Pseudoalgebras to G-G-Categories: 5. Equivariant orthogonal spectra; 6. From G-G-categories to G-G-spaces; 7. Symmetric Gtop-monoidal equivariant K-theory; Appendices: Appendix A. Symmetric monoidal categories; Appendix B. Multicategories; Appendix C. Classifying space; Bibliography; Index.

Hugh L. Montgomery, Robert C. Vaughan

This long-anticipated work shares the aims of its celebrated companion: namely, to provide an introduction for students and a reference for researchers to the techniques, results, and terminology of multiplicative number theory. This volume builds on the earlier one (which served as an introduction to basic, classical results) and focuses on sieve methods. This area has witnessed a number of major advances in recent years, e.g. gaps between primes, large values of Dirichlet polynomials and zero density estimates, all of which feature here. Despite the fact that the book can serve as an entry to contemporary mathematics, it remains largely self-contained, with appendices containing background or material more advanced than undergraduate mathematics. Again, exercises, of which there is a profusion, illustrate the theory or indicate ways in which it can be developed. Each chapter ends with a thorough set of references, which will be essential for all analytic number theorists.

16. Exponential sums I: Van der Corput's method; 17. Estimates for sums over primes; 18. Additive prime number theory; 19. The large sieve; 20. Primes in arithmetic progressions: III; 21. Sieves of fixed dimension and applications; 22. Bounded gaps between primes; E. Topics in harmonic analysis II; F. Uniform distribution; G. Bounds for bilinear forms; H. Linear programming; Errata for Volume I; Name index; Subject index.

Naming New Things and Concepts in Early Modern Science: The Case of Natural History

Naming new discoveries is central to science, and for centuries, Latin dominated this process. The resulting terminology still shapes modern science, yet the influences behind its creation have remained largely unexplored. This is the first comprehensive exploration of how modern scientific terminology took shape during the early modern period. Far from being the product of individual scientists or institutions, the development of this terminology emerged over several centuries, involving a remarkably diverse range of contributors. In particular, the process was often influenced by factors unrelated to science itself – such as the appeal of certain linguistic forms or even sheer coincidence – revealing the unexpected and sometimes arbitrary forces behind the creation of technical terms.

Acknowledgements; Introduction; 1. Neo-Latin as scientific language and the importance of naming; 2. Naming in early modern science; 3. Naming the classes of objects and the disciplines; 4. Natural objects of the old world; 5. Natural objects of the new world; 6. Naming the microscopic world; Conclusion; References; Index.

From social networks to biological systems, networks are a fundamental part of modern life. Network analysis is increasingly popular across the mathematical, physical, life and social sciences, offering insights into a range of phenomena, from developing new drugs based on intracellular interactions, to understanding the influence of social interactions on behaviour patterns. This book provides a toolkit for analyzing random networks, together with theoretical justification of the methods proposed. It combines methods from both probability and statistics, teaching how to build and analyze plausible models for random networks, and how to validate such models, to detect unusual features in the data, and to make predictions. Theoretical results are motivated by applications across a range of fields, and classical data sets are used for illustration throughout the book. This book offers a comprehensive introduction to the field for graduate students and researchers.

1. Introduction; Part I. Basic Setting: 2. Network data sets; 3. Network summaries; 4. Models for networks; Part II. Probability Preliminaries: 5. Branching processes; 6. Some birth and death processes; 7. Poisson approximation; 8. Ramifications of Poisson approximation; 9. Normal approximation; 10. Multivariate normal approximation; Part III. Network Models: 11. The Bernoulli random graph; 12. Models related to the Bernoulli random graph; 13. The Chung–Lu model; 14. The configuration and GPDS models; 15. Random geometric graphs; 16. Small world graphs; 17. Preferential attachment models; 18. Dense graph limits and graphon models; 19. Random processes on networks; 20. Summary of Chapters 5–19; Part IV. Network Inference: 21. Sampling from networks; 22. Estimation: fitting a network model; 23. Assessing model fit; 24. Community detection; 25. Using networks for inference; 26. Some further topics; Appendix; References; Index.

Optimization is a foundational topic in mathematics, underpinning nearly all of our modern industrial and technological world. Assuming only basic knowledge of linear algebra and calculus, this book provides a rapid, yet thorough, overview of applied mathematical optimization for advanced undergraduates, beginning graduate students, or practitioners in science and engineering. The text opens with an “Optimization Bootcamp”, introducing methods at a beginning level, before progressing to deep-dives into advanced topics and research-ready methods. The focus throughout is on modern applications of machine learning, inverse problems, and control. Rich pedagogy includes Python code with simple working examples and advanced case studies. Every section is accompanied by YouTube lectures to encourage interaction with the material. Using intuitive explanations, this book makes the material as simple and interesting as possible, while still having the depth, breadth and precision required to empower use in research and real-world applications.

Preface; Acknowledgments; 1. Optimization bootcamp; 2. Gradient based optimization; 3. Linear programming; 4. Least-squares regression; 5. Nonsmooth and global optimization; 6. Constraints and duality; 7. Bayesian modeling and estimation; 8. Optimization for inverse problems; 9. Optimization for control; 10. Optimization for machine learning; Glossary; Bibliography.

Maths for scientists, SCIENCE / Physics / Mathematical & Computational

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that. Topics covered include correlation inequalities, Lee-Yang theorems, the Peierls method, the Hohenberg-Mermin-Wagner method, infrared bounds, random cluster methods, random current methods and BKT transition. The final section outlines major open problems to inspire future work. This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

Notation; 1. A framework for lattice gases; 2. Correlation inequalities; 3. The Lee-Yang theorem; 4. Peierls argument; 5. No continuous symmetry breaking in 2D (and other situations); 6. Infrared bounds; 7. Stochastic geometric representations; 8. Berezinskii-Kosterlitz-Thouless transition; 9. Final thoughts; References; Author index; Subject index.

Wittgenstein once said that his aim was to make the philosophical problems 'completely disappear', a remark that has baffled philosophers ever since. In this book, Sorin Bangu reconstructs and defends Wittgenstein's unusual idea, and applies it to the traditional problems in philosophy of mathematics, setting out and explaining the subtleties of what is considered the most difficult area of Wittgenstein's views. He also considers how, according to the later Wittgenstein, we should think of the relation between philosophy and mathematics, articulating Wittgenstein's 'normativist' dissolution strategy and explaining his 'therapeutic' vision of the relation between the two disciplines. His book shows how these controversial views sit within the context of current debates in the philosophy of mathematics, and mounts a detailed and convincing defence of the radical eliminative claim – that philosophy of mathematics after Wittgenstein is devoid of its traditional problems.

Introduction: a disappearing act; 1. Wittgenstein's normativism about mathematics; 2. Normativism as eliminativism: three illustrations; 3. Basic mathematical rules (I): hardening; 4. Basic mathematical rules (II): objectivity, agreement, and skepticism; 5. Advanced rules: proofs and concept-formation; 6. Cantor's proof (I): a 'style of thinking'; 7. Cantor's proof (II): reluctant nonrevisionism; 8. Logicism, number and the infinite; 9. Conclusion: a cleared ground; Bibliography.

Wittgenstein once said that his aim was to make the philosophical problems 'completely disappear', a remark that has baffled philosophers ever since. In this book, Sorin Bangu reconstructs and defends Wittgenstein's unusual idea, and applies it to the traditional problems in philosophy of mathematics, setting out and explaining the subtleties of what is considered the most difficult area of Wittgenstein's views. He also considers how, according to the later Wittgenstein, we should think of the relation between philosophy and mathematics, articulating Wittgenstein's 'normativist' dissolution strategy and explaining his 'therapeutic' vision of the relation between the two disciplines. His book shows how these controversial views sit within the context of current debates in the philosophy of mathematics, and mounts a detailed and convincing defence of the radical eliminative claim – that philosophy of mathematics after Wittgenstein is devoid of its traditional problems.

Introduction: a disappearing act; 1. Wittgenstein's normativism about mathematics; 2. Normativism as eliminativism: three illustrations; 3. Basic mathematical rules (I): hardening; 4. Basic mathematical rules (II): objectivity, agreement, and skepticism; 5. Advanced rules: proofs and concept-formation; 6. Cantor's proof (I): a 'style of thinking'; 7. Cantor's proof (II): reluctant nonrevisionism; 8. Logicism, number and the infinite; 9. Conclusion: a cleared ground; Bibliography.

The mathematical method and the nature of mathematical knowledge were subjects of intense philosophical discussion in the 17th and 18th centuries. In particular, there was a debate over whether metaphysical truths admit of distinct proof as geometrical truths do, and whether they may be known with the same degree of certainty. This comparison between geometry and philosophy required a proper understanding of how Euclidean demonstration secured certainty. This element examines attempts by Descartes, Locke, Leibniz, Wolff, Lambert, Mendelssohn and Kant to address this question. The emphasis is on metaphysical and epistemological questions about geometrical demonstration in the 17th- and 18th-centuries.

1. Introduction; 2. Descartes: from the rules to the meditations; 3. Locke; 4. Leibniz; 5. Wolff on the geometrical method; 6. The prize essay competition: Mendelssohn, Lambert and Kant; 7. Kant's critical philosophy of mathematics; 8. Conclusion; Primary sources.

The mathematical method and the nature of mathematical knowledge were subjects of intense philosophical discussion in the 17th and 18th centuries. In particular, there was a debate over whether metaphysical truths admit of distinct proof as geometrical truths do, and whether they may be known with the same degree of certainty. This comparison between geometry and philosophy required a proper understanding of how Euclidean demonstration secured certainty. This element examines attempts by Descartes, Locke, Leibniz, Wolff, Lambert, Mendelssohn and Kant to address this question. The emphasis is on metaphysical and epistemological questions about geometrical demonstration in the 17th- and 18th-centuries.

1. Introduction; 2. Descartes: from the rules to the meditations; 3. Locke; 4. Leibniz; 5. Wolff on the geometrical method; 6. The prize essay competition: Mendelssohn, Lambert and Kant; 7. Kant's critical philosophy of mathematics; 8. Conclusion; Primary sources.

Play of Chance and Purpose: An Invitation to Probability, Statistics, and Stochasticity Using Simulations and Projects

Probability & statistics, MATHEMATICS / Probability & Statistics / General

Play of Chance and Purpose emphasizes learning probability, statistics, and stochasticity by developing intuition and fostering imagination as a pedagogical approach. This book is meant for undergraduate and graduate students of basic sciences, applied sciences, engineering, and social sciences as an introduction to fundamental as well as advanced topics. The text has evolved out of the author's experience of teaching courses on probability, statistics, and stochastic processes at both undergraduate and graduate levels in India and the United States. Readers will get an opportunity to work on several examples from real-life applications and pursue projects and case-study analyses as capstone exercises in each chapter. Many projects involve the development of visual simulations of complex stochastic processes. This will augment the learners' comprehension of the subject and consequently train them to apply their learnings to solve hitherto unseen problems in science and engineering.

List of Tables; List of Figures; Preface; Acknowledgements; Chapter 1. Thinking in Probability; Chapter 2. Probability Distributions; Chapter 3. Discrete Time Markov Chains; Chapter 4. Continuous Time Markov Chains and Queues; Chapter 5. Statistical Experiments; Chapter 6. Prediction and Time Series Modelling; Chapter 7. Glimpses of multivariate statistics; Appendix: A Brief Tour of Matrices; Appendix: A Breeze through MATLAB; Index.

Everywhere one looks, one finds dynamic interacting systems: entities expressing and receiving signals between each other and acting and evolving accordingly over time. In this book, the authors give a new syntax for modeling such systems, describing a mathematical theory of interfaces and the way they connect. The discussion is guided by a rich mathematical structure called the category of polynomial functors. The authors synthesize current knowledge to provide a grounded introduction to the material, starting with set theory and building up to specific cases of category-theoretic concepts such as limits, adjunctions, monoidal products, closures, comonoids, comodules, and bicomodules. The text interleaves rigorous mathematical theory with concrete applications, providing detailed examples illustrated with graphical notation as well as exercises with solutions. Graduate students and scholars from a diverse array of backgrounds will appreciate this common language by which to study interactive systems categorically.

Part I. The Category of Polynomial Functors: 1. Representable functors from the category of sets; 2. Polynomial functors; 3. The category of polynomial functors; 4. Dynamical systems as dependent lenses; 5. More categorical properties of polynomials; Part II. A Different Category of Categories: 6. The composition product; 7. Polynomial comonoids and retrofunctors; 8. Categorical properties of polynomial comonoids; References; Index.

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.

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.

Providing comprehensive yet accessible coverage, this is the first graduate-level textbook dedicated to the mathematical theory of risk measures. It explains how economic and financial principles result in a profound mathematical theory that allows us to quantify risk in monetary terms, giving rise to risk measures. Each chapter is designed to match the length of one or two lectures, covering the core theory in a self-contained manner, with exercises included in every chapter. Additional material sections then provide further background and insights for those looking to delve deeper. This two-layer modular design makes the book suitable as the basis for diverse lecture courses of varying length and level, and a valuable resource for researchers.

Introduction; Acknowledgements; 1. Gains, quantiles and Value-at-Risk; 2. Monetary property and acceptance sets; 3. Diversification, convexity and coherence; 4. Average-Value-at-Risk; 5. Dual representation of convex and coherent risk measures; 6. Representation theorems for risk measures on $L^p$-spaces; 7. Constructions of risk measures; 8. Law-determined risk measures; 9. Law-determined risk measures on $L^p$-spaces; 10. Comonotonicity and Choquet integrals; 11. Coherent comonotonic additive risk measures; 12. Multivariate risk measures; List of representations of coherent risk measures; List of important law-determined risk measures; References; Index.

Providing comprehensive yet accessible coverage, this is the first graduate-level textbook dedicated to the mathematical theory of risk measures. It explains how economic and financial principles result in a profound mathematical theory that allows us to quantify risk in monetary terms, giving rise to risk measures. Each chapter is designed to match the length of one or two lectures, covering the core theory in a self-contained manner, with exercises included in every chapter. Additional material sections then provide further background and insights for those looking to delve deeper. This two-layer modular design makes the book suitable as the basis for diverse lecture courses of varying length and level, and a valuable resource for researchers.

Introduction; Acknowledgements; 1. Gains, quantiles and Value-at-Risk; 2. Monetary property and acceptance sets; 3. Diversification, convexity and coherence; 4. Average-Value-at-Risk; 5. Dual representation of convex and coherent risk measures; 6. Representation theorems for risk measures on $L^p$-spaces; 7. Constructions of risk measures; 8. Law-determined risk measures; 9. Law-determined risk measures on $L^p$-spaces; 10. Comonotonicity and Choquet integrals; 11. Coherent comonotonic additive risk measures; 12. Multivariate risk measures; List of representations of coherent risk measures; List of important law-determined risk measures; References; Index.

Historically and conceptually, influential traditions of thought and practice associated with humanism and science have been deeply connected. This book explores some of the most pivotal relations of humanistic and scientific engagement with the world to inspire a reconsideration of them in the present. Collectively, its essays illuminate a fundamental but contested feature of a broadly humanist worldview: the hope that science may help to improve the human condition, as well as the myriad relationships of humanity to the natural and social worlds in which we live. Arguably, these relationships are now more profoundly interwoven with our sciences and technologies than ever before. Addressing scientific and other forms of inquiry, approaches to integrating humanism with science, and cases in which science has failed, succeeded, and could do more to promote our collective welfare, this book enjoins us to articulate a compelling, humanist conception of the sciences for our times. This title is also available as Open Access on Cambridge Core.

Introduction Anjan Chakravartty; Part I. Interrelations: Scientific and Other Forms of Knowledge: 1. What is science for? Modern intersections of science and humanism Anjan Chakravartty; 2. Varieties of philosophical humanism and conceptions of science Ian James Kidd; 3. Scientism and the limits of objectivity Gurpreet Rattan; 4. Scientism: reflections on nature, value, and agency Akeel Bilgrami; Part II. Inspirations: Philosophies of Science and its Social Role: 5. Scientific humanisms: Sarton, Reichenbach, and the cultural crisis of western science after world war one Alan Richardson; 6. John Dewey, humanism, and the value of science Aleksandra Hernandez; 7. Socio-political engagement and scientific value freedom: the view from the left vienna circle Thomas Uebel; 8. The pragmatic and the religious functions of science Matthew J. Brown; Part III. Interventions: Scientific Knowledge and Social Imperatives: 9. The present plight of science, and our plight Janet A. Kourany; 10. Science and justice: beyond the new orthodoxy of value-laden science David Ludwig; 11. The human sciences and the 'theory of women' Catherine Wilson; 12. Toward more inclusive research: new challenges and responsibilities for scientists, philosophers, and citizens Stéphanie Ruphy; References; Index.

Over the last couple of decades, there has been increasing concern about the alleged rise of various forms of science denial. But what exactly is science denial? Is it really on the rise? If so, what explains its rise? And what is so concerning about it? This Element argues that the notion of science denial is highly ambiguous and that, once we carefully distinguish among all the different phenomena that are often conflated under this label, it is doubtful that any of them warrants all of the concerns that animate the critics of science denial. This has important consequences for how we understand the complex and delicate relationship between science and the public and, more generally, the collective epistemic malaise afflicting liberal democracies.

1. Introduction; 2. Why is science denial concerning?; 3. What Is science denial?; 4. What explains science denial?; 5. Conclusion: beyond 'science denial'; References.

Over the last couple of decades, there has been increasing concern about the alleged rise of various forms of science denial. But what exactly is science denial? Is it really on the rise? If so, what explains its rise? And what is so concerning about it? This Element argues that the notion of science denial is highly ambiguous and that, once we carefully distinguish among all the different phenomena that are often conflated under this label, it is doubtful that any of them warrants all of the concerns that animate the critics of science denial. This has important consequences for how we understand the complex and delicate relationship between science and the public and, more generally, the collective epistemic malaise afflicting liberal democracies.

1. Introduction; 2. Why is science denial concerning?; 3. What Is science denial?; 4. What explains science denial?; 5. Conclusion: beyond 'science denial'; References.

This two-part book offers a rigorous yet accessible exploration of set theory and transfinite algebra, with a particular emphasis on the axiom of choice and its applications. Part I presents an informal axiomatic introduction to the foundations of set theory, including a detailed treatment of the axiom of choice and its equivalents, suitable for advanced undergraduates. Part II, aimed at graduate students and professional mathematicians, treats selected topics in transfinite algebra where the axiom of choice, in one form or another, is useful or even indispensable. The text features self-contained chapters for flexible use, and includes material rarely found in the literature, such as Tarski's work on complete lattices, Hamel's solution to Cauchy's functional equation, and Artin's resolution of Hilbert's 17th problem. Over 140 exercises, with full solutions provided in the Appendix, support active engagement and deeper understanding, making this a valuable resource for both independent study and course preparation.

Introduction; Part I. Set Theory: 1. The axioms of set theory; 2. Correspondences, mappings, and quotient sets; 3. Ordered sets; 4. Around the axiom of choice; 5. Cardinals and ordinals; 6. First-order logic and the axioms of set theory revisited; 7. Some excursions; Part II. Topics in Transfinite Algebra: 8. Group and ring structures on non-empty sets; 9. Orderable abelian groups and fields; 10. Subdirect decomposition of algebras; 11. Dependence relations, rank functions, and closure operators; 12. Semisimple and injective modules; 13. The Jacobson radical of a ring; 14. Artin's solution of Hilbert's 17th problem; Appendix. Solutions to exercises; References; Index.

This two-part book offers a rigorous yet accessible exploration of set theory and transfinite algebra, with a particular emphasis on the axiom of choice and its applications. Part I presents an informal axiomatic introduction to the foundations of set theory, including a detailed treatment of the axiom of choice and its equivalents, suitable for advanced undergraduates. Part II, aimed at graduate students and professional mathematicians, treats selected topics in transfinite algebra where the axiom of choice, in one form or another, is useful or even indispensable. The text features self-contained chapters for flexible use, and includes material rarely found in the literature, such as Tarski's work on complete lattices, Hamel's solution to Cauchy's functional equation, and Artin's resolution of Hilbert's 17th problem. Over 140 exercises, with full solutions provided in the Appendix, support active engagement and deeper understanding, making this a valuable resource for both independent study and course preparation.

Introduction; Part I. Set Theory: 1. The axioms of set theory; 2. Correspondences, mappings, and quotient sets; 3. Ordered sets; 4. Around the axiom of choice; 5. Cardinals and ordinals; 6. First-order logic and the axioms of set theory revisited; 7. Some excursions; Part II. Topics in Transfinite Algebra: 8. Group and ring structures on non-empty sets; 9. Orderable abelian groups and fields; 10. Subdirect decomposition of algebras; 11. Dependence relations, rank functions, and closure operators; 12. Semisimple and injective modules; 13. The Jacobson radical of a ring; 14. Artin's solution of Hilbert's 17th problem; Appendix. Solutions to exercises; References; Index.

Alberto Debernardi Pinos, Elijah Liflyand, Maria Zeltser, Sergey Tikhonov

Addressing a significant gap in the study of number series, this book presents an in-depth theory of multiple number series and an exhaustive examination of one-dimensional series. It incorporates overlooked yet essential results alongside recent research advancements. Much of the text is based on the authors' original contributions, particularly in the development of relaxed monotonicity concepts, which have become fundamental tools in Fourier and functional analysis. Each chapter concludes with historical context, aiding readers in understanding the theory's evolution. The book is aimed at a wide audience, ranging from undergraduate students to experts in the field. It offers a modern perspective on the theory, along with detailed introductory chapters that make complex concepts accessible for students. The audience will find the novel contributions enriching and inspiring.

Introduction; Part I. Single Series: 1. Generalized monotonicity; 2. Number series in dimension one; 3. Miscellaneous tests and results; Part II. Double Series: 4. Rudiments of double sequences and series; 5. Tests for convergence of double series; 6. Convergence tests involving generalizations of monotonicity; Part III. Multiple Series: 7. Rudiments of multiple sequences and series; 8. Tests for convergence of multiple series; 9. Multiple series with relaxed monotonicity; Table of correspondences; References – Monographs and Textbooks; References; Symbol index; Index.

Dong Zhang, Jürgen Jost, Raffaella Mulas

Addressing the active and challenging field of spectral theory, this book develops the general theory of spectra of discrete structures, on graphs, simplicial complexes, and hypergraphs. In fact, hypergraphs have long been neglected in mathematical research, but because of the discovery of Laplace operators that can probe their structure, and their manifold applications from chemical reaction networks to social interactions, they have now become one of the most active areas of interdisciplinary research. The authors' analysis of spectra of discrete structures embeds intuitive and easily visualized examples, which are often quite subtle, within a general mathematical framework. They highlight novel research on Cheeger-type inequalities that connect spectral estimates with the geometry, more precisely the cohesion, of the underlying structure. Establishing mathematical foundations and demonstrating applications, this book will be of interest to graduate students and researchers in mathematics working on the spectral theory of operators on discrete structures.

Preface; Acknowledgments; Organization of the book; Part I. Basics: Foundational Material, Elementary Aspects, Examples: 1. Introduction; 2. The abstract setting; 3. The classical case; 4. First properties of the spectrum; 5. Floer theory; Part II. Eigenvalues and Eigenfunctions on Simplicial Complexes and Hypergraphs: 6. Lovász extensions; 7. Discrete p-Laplacians; 8. Cheeger inequalities; 9. Nodal domains; Part III. Additional topics: Interlacing, Tensors, Non-backtracking Laplacians, and Applications: 10. Interlacing and spectral classes; 11. Spectral theory of hypergraphs via tensors; 12. The non-backtracking Laplacian; 13. Applications; Bibliography; Index.

This monograph extends the classical spectral theory of ordinary graphs to the broader framework of signed graphs. It integrates foundational results with recent advances, explores applications, and clarifies connections with related mathematical structures while indicating promising directions for future research. The exposition remains rigorous throughout, presenting core concepts, major developments, and emerging ideas in a coherent and accessible manner. Complementing the theoretical material, the monograph includes illustrative examples and problem sections to support understanding and encourage continued study. This monograph will serve as a reference for mathematicians working in the spectral theory of signed graphs as well as a tutorial for graduate students entering the subject area and computer scientists, chemists, physicists, biologists, electrical engineers and others whose work involves graph-based modelling.

Preface; 1. Introduction; 2. Eigenvalues; 3. Strong regularity; 4. Star complements; 5. Signed line graphs; 6. Signed graphs with a small number of eigenvalues; 7. Integrality and cospectrality; 8. Polynomial reconstruction; 9. Related structures; 10. Generalizations; References; Index.

Andrei Gagarin, Iskander Aliev, Matthew Lettington, Roger Behrend

This volume contains eight survey articles by the invited speakers of the 31st British Combinatorial Conference, held at Cardiff University in July 2026. Each article provides an overview of recent developments in a current hot research topic in combinatorics. Topics covered include random planar graphs, temporal graphs, domino tilings, extremal poset theory, asymptotic enumeration, graph homomorphisms, combinatorial rigidity theory, logic and model theory, matroids, and graph bootstrap percolation. The authors are among the world's foremost researchers on their respective topics, but their surveys are accessible to nonspecialist readers: they are written clearly, with little prior knowledge assumed, and with pointers to the wider literature. Taken together, these surveys give a snapshot of the research frontier in contemporary combinatorics, helping researchers and graduate students in mathematics and theoretical computer science to keep abreast of the latest developments in the field.

1. Phase Transitions in Random Planar Graphs Mihyun Kang; 2. Extremal Poset Theory Maria Axenovich, Ryan R. Martin and Balázs Patkós; 3. Techniques for Taming Temporal Graphs Kitty Meeks; 4. Asymptotic Enumeration via Graph Containers and Entropy Jinyoung Park; 5. Graph Bootstrap Percolation – A Discovery of Slowness David Fabian, Patrick Morris and Tibor Szabó; 6. Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach James Cruickshank, Bill Jackson, Tibor Jordán and Shin-ichi Tanigawa; 7. Monadic Second-Order Logic for Hypergraphs and Matroids Dillon Mayhew; 8. Domino Tilings Beyond 2D Caroline J. Klivans and Nicolau Saldanha.

The 'arrow of time,' a concept first introduced by Sir Arthur Eddington, reflects the one-way flow of time and its association with various physical asymmetries in thermodynamics, cosmology, quantum mechanics, field theories, and beyond. Yet, the foundations of the arrow of time continues to challenge physicists and philosophers, having profound implications across multiple theories and disciplines.This volume compiles insights from the international colloquium 'The Arrow of Time: From Local Systems to the Whole Univers' held in Buenos Aires in 2023. It explores diverse perspectives on the arrow of time in thermodynamics, quantum mechanics and cosmology, its relation to counterfactual reasoning, free will and the growing-block universe, the interplay between consciousness and time, and the implications of time-reversal invariance. Collectively, these contributions provide a rigorous and comprehensive analysis of the enduring enigma of time's unidirectional nature.

List of contributors; Preface; Introduction; Part I. Local Systems: 1. The direction of time: from the cosmos to local systems David Albert; 2. Thermodynamics with and without irreversibility David Wallace; 3. Thermodynamics without time Francesca Vidotto; Part II. The Arrow of Time and Philosophical Issues: 4. Explaining the psychological arrow of time in time-symmetric physics Meir Hemmo and Orly Shenker; 5. The consequence argument meets the mentaculus Barry Loewer; 6. 'Freedom from inside out'. Revisited Carl Hoefer; 7. The arrow of time is alive and well but forbidden under the received view in physics Ruth E. Kastner; Part III. The Arrow of Time and Time-Reversal Invariance: 8. Time for pancakes: ontology and time reversal Valia Allori; 9. Insights into quantum time reversal from the classical Schrödinger equation Craig Callender; 10. Saving the appearances: the direction of time and time-reversal invariance Cristian López; Part IV. The Whole Universe: 11. Following Earman's time direction heresy: from the global arrow of time to local irreversible processes Olimpia Lombardi; 12. The seeds of cosmic structure, the black hole information puzzle and the entropic arrow of time Daniel Sudarsky and Elías Okón; 13. Finding time for Wheeler-DeWitt cosmology Nick Huggett , Karim Thébault; Index.

When you see a paper crane, what do you think of? A symbol of hope, a delicate craft, The Karate Kid? What you might not see, but is ever present, is the fascinating mathematics underlying it. Origami is increasingly applied to engineering problems, including origami-based stents, deployment of solar arrays in space, architecture, and even furniture design. The topic is actively developing, with recent discoveries at the frontier (e.g., in rigid origami and in curved-crease origami) and an infusion of techniques and algorithms from theoretical computer science. The mathematics is often advanced, but this book instead relies on geometric intuition, making it accessible to readers with only a high school geometry and trigonometry background. Through careful exposition, more than 160 color figures, and 49 exercises all completely solved in an Appendix, the beautiful mathematics leading to stunning origami designs can be appreciated by students, teachers, engineers, and artists alike.

Preface; 1. Introduction; 2. Stamp folding; 3. Flat vertex folds; 4. Flat folding is hard; 5. Rigid origami and degree-4 vertices; 6. Origami design; 7. Fold & 1-cut; 8. Curved crease origami; 9. Self-folding origami; 10. Origamizer; Appendix A. Beyond: topics not covered; Appendix B. Solutions to exercises; References; Index.

When you see a paper crane, what do you think of? A symbol of hope, a delicate craft, The Karate Kid? What you might not see, but is ever present, is the fascinating mathematics underlying it. Origami is increasingly applied to engineering problems, including origami-based stents, deployment of solar arrays in space, architecture, and even furniture design. The topic is actively developing, with recent discoveries at the frontier (e.g., in rigid origami and in curved-crease origami) and an infusion of techniques and algorithms from theoretical computer science. The mathematics is often advanced, but this book instead relies on geometric intuition, making it accessible to readers with only a high school geometry and trigonometry background. Through careful exposition, more than 160 color figures, and 49 exercises all completely solved in an Appendix, the beautiful mathematics leading to stunning origami designs can be appreciated by students, teachers, engineers, and artists alike.

Preface; 1. Introduction; 2. Stamp folding; 3. Flat vertex folds; 4. Flat folding is hard; 5. Rigid origami and degree-4 vertices; 6. Origami design; 7. Fold & 1-cut; 8. Curved crease origami; 9. Self-folding origami; 10. Origamizer; Appendix A. Beyond: topics not covered; Appendix B. Solutions to exercises; References; Index.

Ambat Vijayakumar, Aparna Lakshmanan S., Peter J. Cameron

The Shrikhande graph, discovered by Indian Mathematician Sharadchandra Shankar Shrikhande in 1959, exhibits several unusual properties and occupies a pivotal position within discrete mathematics. Offering a unique introduction to graph theory and discrete mathematics, this book uses the example of the Shrikhande graph as a window through which these topics can be explored. Providing historical background, including the Euler conjecture and its demise, the authors explore key concepts including: Cayley graphs; topological graph theory; spectral theory; Latin squares; root systems. A novel and valuable resource for graduate students and researchers interested in graph theory, its history, and applications, this book offers a comprehensive exploration of the Shrikhande graph and its significance.

Preface; Part I. Biography: 1. The life of S. S. Shrikhande; Part II. Graph Basics: 2. Definitions; 3. Strongly regular graphs; Part III. Properties of the Shrikhand Graph:. 4. Spectrum and automorphism group; 5. Further properties; Part IV. The Shrikhande Graph in Context: 6. Latin squares; 7. The Shrikhande graph on the torus; 8. Root systems; 9. Graphs with least eigenvalue; 10. Miscellanea; 11. Further reading; References; Index.

Ambat Vijayakumar, Aparna Lakshmanan S., Peter J. Cameron

The Shrikhande graph, discovered by Indian Mathematician Sharadchandra Shankar Shrikhande in 1959, exhibits several unusual properties and occupies a pivotal position within discrete mathematics. Offering a unique introduction to graph theory and discrete mathematics, this book uses the example of the Shrikhande graph as a window through which these topics can be explored. Providing historical background, including the Euler conjecture and its demise, the authors explore key concepts including: Cayley graphs; topological graph theory; spectral theory; Latin squares; root systems. A novel and valuable resource for graduate students and researchers interested in graph theory, its history, and applications, this book offers a comprehensive exploration of the Shrikhande graph and its significance.

Preface; Part I. Biography: 1. The life of S. S. Shrikhande; Part II. Graph Basics: 2. Definitions; 3. Strongly regular graphs; Part III. Properties of the Shrikhand Graph:. 4. Spectrum and automorphism group; 5. Further properties; Part IV. The Shrikhande Graph in Context: 6. Latin squares; 7. The Shrikhande graph on the torus; 8. Root systems; 9. Graphs with least eigenvalue; 10. Miscellanea; 11. Further reading; References; Index.

Carlos Tomei, Guillaume Dubach, Percy Deift, Thomas Trogdon

The Toda Lattice and Universality for the Computation of the Eigenvalues of a Random Matrix

Written by leaders in the field, this text showcases some of the remarkable properties of the finite Toda lattice and applies this theory to establish universality for the associated Toda eigenvalue algorithm for random Hermitian matrices. The authors expand on a 2019 course at the Courant Institute to provide a comprehensive introduction to the area, including previously unpublished results. They begin with a brief overview of Hamiltonian mechanics and symplectic manifolds, then derive the action-angle variables for the Toda lattice on symmetric matrices. This text is one of the first to feature a new perspective on the Toda lattice that does not use the Hamiltonian structure to analyze its dynamics. Finally, portions of the above theory are combined with random matrix theory to establish universality for the runtime of the associated Toda algorithm for eigenvalue computation.

1. Introduction; 2. Hamiltonian mechanics and integrable systems; 3. The Toda lattice; 4. Toda without Hamiltonian structure; 5. Random matrix ensembles; 6. Universality for the Toda algorithm; References; Notation and Abbreviations; Index.

Spanning elementary, algebraic, and analytic approaches, this book provides an introductory overview of essential themes in number theory. Designed for mathematics students, it progresses from undergraduate-accessible material requiring only basic abstract algebra to graduate-level topics demanding familiarity with algebra and complex analysis. The first part covers classical themes: congruences, quadratic reciprocity, partitions, cryptographic applications, and continued fractions with connections to quadratic Diophantine equations. The second part introduces key algebraic tools, including Noetherian and Dedekind rings, then develops the finiteness of class groups in number fields and the analytic class number formula. It also examines quadratic fields and binary quadratic forms, presenting reduction theory for both definite and indefinite cases. The final section focuses on analytic methods: L-series, primes in arithmetic progressions, and the Riemann zeta function. It addresses the Prime Number Theorem and explicit formulas of von Mangoldt and Riemann, equipping students with foundational knowledge across number theory's major branches.

Preface; Part I. Elementary Methods: 1. Congruences and primes; 2. Continued fractions; 3. Euclidean and principal ideal domains; Part II. Algebraic Methods: 4. Some commutative algebra; 5. Integrality; 6. Ideal class groups and units; 7. Quadratic fields and binary quadratic forms; 8. Cyclotomic fields; Part III. Analytic Methods: 9. Dirichlet series; 10. The Riemann zeta function; 11. The prime number theorem; Bibliography; Index.

Filling a gap in the literature, this book explores the theory of gradient flows of convex functionals in metric measure spaces, with an emphasis on weak solutions. It is largely self-contained and assumes only a basic understanding of functional analysis and partial differential equations. With appendices on convex analysis and the basics of analysis in metric spaces, it provides a clear introduction to the topic for graduate students and non-specialist researchers, and a useful reference for anyone working in analysis and PDEs. The text focuses on several key recent developments and advances in the field, paying careful attention to technical detail. These include how to use a first-order differential structure to construct weak solutions to the p-Laplacian evolution equation and the total variation flow in metric spaces, how to show a Euler–Lagrange characterisation of least gradient functions in this setting, and how to study metric counterparts of Cheeger problems.

1. Analysis in metric spaces; 2. The p-Laplacian evolution equation; 3. Gradient flows of functionals with inhomogeneous growth; 4. A general Gauss-Green formula; 5. Total variation flow on the whole space; 6. Total variation flow on bounded domains; 7. Applications to related problems; Appendix A. Results from Convex Analysis; Appendix B. Equivalent definitions of Sobolev and BV spaces.