Böcker i Cambridge Studies in Advanced Mathematics-serien

A friendly introduction to higher index theory, a rapidly-developing subject at the intersection of geometry, topology and operator algebras. A well-balanced combination of introductory material (with exercises), cutting-edge developments and references to the wider literature make this book a valuable guide for graduate students and experts alike.

The character theory of reductive groups over finite fields is a rich, complex and vast field, incorporating tools and methods from many mathematical areas. This book provides a contemporary treatment of the theory and is a useful reference for graduate students and researchers with a basic understanding of algebraic geometry.

This comprehensive introduction to stable homotopy theory presents the foundations of this often daunting subject together in one place for the first time. Writing with beginning graduate students in mind, the authors begin with the motivating phenomena before discussing the general theory and moving on to current research and applications.

Recent years have seen a flurry of advances in Fourier restriction theory, with significant applications in number theory, PDEs and geometric analysis. This timely text for specialists and graduate students in analysis, written by a leader in the field, brings the reader to the forefront of these exciting developments.

This book covers the spectral theory of Toeplitz forms and Wiener-Hopf integral operators and their applications in harmonic and functional analysis. With exercises and solutions, historical background, and surveys of the latest results, this is an essential source on Toeplitz theory for students and researchers alike.

This comprehensive introduction to sub-Riemannian geometry proceeds from classical topics to cutting-edge theory and applications. The only prerequisites are calculus, linear algebra and differential equations. It can be used for graduate courses in Riemannian or sub-Riemannian geometry, or as a reference for researchers in several disciplines.

This book is the first systematic exposition of the theory of derived categories. It carefully explains the foundations before moving on to key applications in (non)commutative algebra, including derived categories of DG modules. Many examples and exercises serve to demystify this difficult but important part of modern homological algebra.

This user-friendly book introduces modern homotopy theory through the lens of higher categories after Joyal and Lurie. Starting from scratch it guides graduate students and researchers through the powerful tools that the theory provides for applications in such areas as algebraic geometry, representation theory, algebra and logic.

Designed for beginning graduate students, this book introduces and develops the classical results on Hardy spaces and applies them to fundamental problems in modern analysis. With solved exercises, short surveys of recent developments, and engaging accounts of the field's main contributors, this book is the ideal source on Hardy spaces.

Over the last 45 years, Boolean theorem has been generalized and extended in several different directions and its applications have reached into almost every area of modern mathematics. Dr Johnstone begins by developing the theory of locales. This development culminates in the proof of Stone's Representation Theorem.

This is a modern introduction to the analytic techniques used in the investigation of zeta functions, through the example of the Riemann zeta function. Graduate students and non-specialists will find this an up-to-date and accessible introduction.

A new era in the representation theory of finite groups began with the McKay conjecture. This book presents the conjecture in a clear and accessible way, with only minimal pre-requisites. Many fascinating aspects of character theory are explored along the way, with some new and elegant proofs.

This book gets to the heart of the combinatorics that binds together quantum field theory and probability with a unified framework for Wick (normal) ordering and its applications. Featuring many worked examples, it is for mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.

The Erdos-Ko-Rado Theorem is a fundamental result in combinatorics. Aimed at graduate students and researchers, this comprehensive text shows how tools from algebraic graph theory can be applied to prove the EKR Theorem and its generalizations. Readers can test their understanding at every step with the end-of-chapter exercises.

A comprehensive introduction to algebraic geometry and representation theory written by a leading expert in the field. For graduate students and researchers in computer science and mathematics, the book demonstrates state-of-the-art techniques to solve real world problems, focusing on P vs NP and the complexity of matrix multiplication.

Fully illustrated and rigorous in its approach, this is a comprehensive account of geometric techniques for studying the topology of smooth manifolds. Little prior knowledge is assumed, giving advanced students and researchers an accessible route into the wide-ranging field of differential topology.

Martingales arise in many areas of probability theory. This book focuses on their applications to the geometry of Banach spaces and discusses the interplay of Banach space valued martingales with various other areas of analysis. It is accessible to graduates with a basic knowledge of real and complex analysis.

Branching Brownian motion is a key model at the crossroads of value statistics for Gaussian processes, statistical physics, and non-linear partial differential equations. This book gives a concise introduction for graduate students and researchers leading up to the most recent developments in this active area of research.

The study of model spaces is a broad field with connections to complex analysis, operator theory, engineering and mathematical physics. This self-contained text is the ideal introduction for newcomers, quickly taking them through the history of the subject and then pointing towards areas of future research.

Blending theory and applications, this book is a vital resource for graduates and researchers. It offers a broad theoretic base, synthesising symplectic geometry and optimal control theory, essential for mechanical, geometric or space engineering problems. The theory is tested through challenging problems and is rich with fresh insights and ideas.

During the past two decades there has been active interplay between geometric measure theory and Fourier analysis. This book for graduate students and researchers describes part of that development, concentrating on the relationship between the Fourier transform and Hausdorff dimension. It covers classical results as well as cutting-edge research.

Rich with examples, applications and over 400 exercises, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. It requires few prerequisites, beginning with elementary material suitable for undergraduate students and gradually building up to more sophisticated topics.

This book, first published in 2000, is a concise introduction to ring theory, module theory and number theory, ideal for a first year graduate student, as well as an excellent reference for working mathematicians in other areas. About 200 exercises complement the text and introduce further topics.

This graduate-level text demonstrates the basic techniques and how to apply them to various areas of research in geometric analysis. The author focuses mainly on the interaction of partial differential equations with differential geometry and only a rudimentary knowledge of Riemannian geometry and partial differential equations is required.

This engaging graduate-level introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. Explanatory pictures, detailed proofs, exercises and helpful remarks make it suitable for self-study and also a useful reference for researchers.

On its original publication, this algebraic introduction to Grothendieck's local cohomology theory was the first book devoted solely to the topic and it has since become the standard reference for graduate students. This second edition has been thoroughly revised and updated to incorporate recent developments in the field.

The Korteweg-de Vries (KdV), the AKNS, the nonlinear Schroedinger, the sine-Gordon and the Camassa-Holm equations, and the Thirring system, are all completely integrable nonlinear PDEs permitting special classes of solutions. This is a detailed treatment of the class of algebro-geometric solutions and their representations in terms of Riemann theta functions.

This book, first published in 2000, is a concise introduction to ring theory, module theory and number theory, ideal for a first year graduate student, as well as an excellent reference for working mathematicians in other areas. About 200 exercises complement the text and introduce further topics.

This modern approach to the theory of automorphic representations keeps definitions to a minimum, focusing instead on providing concrete examples and detailed proofs of the key theorems. This book is the perfect introduction for students at the advanced undergraduate level and beyond, and for researchers new to the field.

This self-contained treatment of the theory of non-Archimedean locally convex spaces provides a clear exposition of the theory, together with complete proofs. A guide to examples and glossary of terms make the book easily accessible to beginners at the graduate level as well as specialists from various disciplines.