Böcker i Cambridge Studies in Advanced Mathematics-serien

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.

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.

Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderon, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

This work consists of two courses on the moduli spaces of vector bundles. The first is introductory, and assumes very little background; the second is more advanced and takes the reader into current areas of research. This a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.

Arising from a summer school course, this book develops the modern theory of rational varieties at a level that will particularly suit graduate students. The authors have written a state-of-the-art treatment with numerous exercises and solutions to help students reach the stage where they can begin to tackle contemporary research.

This book gives a quick introduction to the theory of foliations and Lie groupoids. It is based on the authors' extensive teaching experience and contains numerous examples and exercises making it ideal either for independent study or as the basis of a graduate course.

This book, first published in 2000, emphasizes the substantial role played by classical complex analysis in understanding holomorphic dynamics and gives an up-to-date coverage of the modern theory. The authors cover entire functions, Kleinian groups and polynomial automorphisms of several complex variables, as well as the case of rational functions.

The first rigorous treatment of tolerance graphs in book form. This book will act as a springboard for researchers, especially graduate students, to pursue new directions of investigation. With many examples and exercises it is also suitable for use as the text for a graduate course in graph theory.

The theory of automorphic forms is a cornerstone in modern number theory, an essential ingredient of the proof of Fermat's Last Theorem. This book presents the subject in a style which is accessible to graduate students entering the field, yet includes topics on the research frontier.

The first text devoted to the theory of natural dualities, this book provides an efficient path through a large body of results, examples and applications in this subject which is otherwise available only in scattered research papers. It is aimed at algebraists, category theorists and those who use algebra in their work, particularly those interested in non-classical logics.

This is the second of two volumes providing an introduction to modern developments in the representation theory of finite groups and associative algebras. Much of the material presented here has never appeared before in book form.