Bücher der Reihe Cambridge Studies in Advanced Mathematics

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.

This is the first book to deal with the new and expanding field of fusion systems, of increasing importance both to algebraists and topologists. Designed for use as a text and reference work, it is suitable for graduate students and experts alike.

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 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.

This concise treatment covers many topics that are central to the subject, but missing from existing textbooks. It contains numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

This stimulating introduction to zeta (and related) functions of graphs develops the fruitful analogy between combinatorics and number theory - for example, the Riemann hypothesis for graphs - making connections with quantum chaos, random matrix theory, and computer science. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

Assuming little mathematical background, this short introduction to category theory is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. Suitable for independent study or as a course book, it gives extensive explanations of the key concepts along with hundreds of examples and exercises.

The theory of random matrices plays an important role in many areas of pure mathematics. This rigorous introduction is specifically designed for graduate students in mathematics or related sciences, who have a background in probability theory but have not been exposed to advanced notions of functional analysis, algebra or geometry.

This informal introduction focuses on the branch of ergodic theory known as isomorphism theory. Exercises, open problems, and helpful hints actively engage the reader and encourage them to participate in developing proofs independently. Ideal for graduate courses, this book is also a valuable reference for the professional mathematician.

The logarithmic integral is a thread connecting many apparently separate parts of twentieth century analysis, and so is a natural point at which to begin a serious study of real and complex analysis. Professor Koosis' aim is to show how, from simple ideas, one can build up an investigation which explains and clarifies many different, seemingly unrelated problems.

A unique development of these two subjects contained in a single volume. New topics featured in this fully revised edition include regular variation and subexponential distributions, characterisation of Levy processes with finite variation, multiple Wiener-Levy integrals and chaos decomposition, and introductions to Malliavin calculus and stability theory for Levy-driven SDEs.

A detailed treatment of the class of algebro-geometric solutions and their representations in terms of Riemann theta functions. Provides a rigorous and self-contained presentation at graduate level. Background material is succinctly presented in four appendices, and detailed notes and an exhaustive bibliography enhance understanding of the main results.

A self-contained and elementary presentation of Lie group theory, concentrating on analysis on Lie groups. The author describes in detail many interesting examples with topics ranging from Haar measure to harmonic functions. With numerous exercises and worked examples, it's ideal for a graduate course on analysis on Lie groups.

A random walk is one of the most studied topics in probability theory and has many important applications outside of mathematics. Ideal for graduate students, this text develops the theory from basic definitions through to current research problems. It is also suitable for mathematicians working in related fields.

Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed, every aspect of the topic is covered, and exercises are included at the end of each chapter.

This authoritative text presents a broad view of the spectral theory of non-self-adjoint linear operators and contains many illustrative examples and exercises. Topics discussed include Fredholm theory, Hilbert-Schmidt and trace class operators, one-parameter semigroups, perturbations of their generators and a thorough account of the new theory of pseudospectra.

Local geometric Langlands Correspondence is a new area of mathematical physics developed by the author and others in the last 20 years. This text for advanced undergraduate students and graduate students, builds the theory from scratch, defines all necessary concepts and proves the essential results along the way.

By the author of Levy Processes, the first comprehensive theoretical account of mathematical models for random and repeated fragmentation and coagulation over time. Written for readers with a solid background in probability, its careful exposition allows graduate students, as well as working mathematicians, to approach the material with confidence.

Now in paperback, this graduate-level textbook introduces the representation theory of semi-simple Lie groups. Containing appendices sketching some basic topics with a comprehensive guide to further reading, it is suitable for research students in algebra and analysis, and for research mathematicians requiring a readable account of the topic.

An introduction to the theory of partial differential operators. For those with a knowledge of introductory functional analysis, up to the spectral theorem for bounded linear operators on Banach spaces. Suitable for self-study or as a course text.

A self-contained introduction to the classical theory and its generalizations, aimed at mathematicians and scientists working in dynamical systems.

This text presents some of the key results in the representation theory of finite groups, and contains ample material for a one semester course. Exercises are provided at the end of most sections; the results of some are used later in the text.

This is an elementary and self-contained introduction to nonlinear functional analysis and its applications, especially in bifurcation theory. The book is intended to accompany advanced courses for students of pure and applied mathematics and physics; exercises are consequently included.

Now in a paperback edition for the first time, this second edition of Some Random Series of Functions covers random series in Banach and Hilbert spaces, random Taylor or Fourier series, Brownian motion and other Gaussian processes, plus certain types of random sets and measures.

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. Detailed references are provided and each section concludes with exercises.

Now in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean space. Examples to which this theory applies include fractal-type objects such as strange attractors for dynamical systems, and those fractals used as models in the sciences.

This book is about the basis of mathematical reasoning both in pure mathematics itself (particularly algebra and topology) and in computer science (how and what it means to prove correctness of programmes). It deliberately transcends disciplinary boundaries and challenges many established attitudes to the foundations of mathematics.