Bücher der Reihe Cambridge Studies in Advanced Mathematics

This book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is a suitable introduction for any mathematician (especially graduate students) to this field.

Category theory structures the mathematical world and is seen everywhere in modern mathematics. This book, suitable for graduate students or researchers with a background in algebraic topology and algebra, provides a self-contained introduction to the theory and explains its important applications to homotopy theory.

This text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject 'elliptic cohomology' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten's genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.

The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.

This is a self-contained, modern treatment of the algebraic theory of machines. Dr Holcombe examines various applications of the idea of a machine in biology, biochemistry and computer science and gives also a rigorous treatment of the way in which these machines can be decomposed and simulated by simpler ones. This treatment is based on fundamental ideas from modern algebra. Motivation for many of the newer results is provided by way of applications so this account should be accessible and valuable for those studying applied algebra or theoretical computer science at advanced undergraduate or beginning postgraduate level, as well as for those undertaking research in those areas.

The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently and to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more, including chapters on discrete orthogonal polynomials and elliptic functions. The authors show how a very large part of the subject traces back to two equations - the hypergeometric equation and the confluent hypergeometric equation - and describe the various ways in which these equations are canonical and special. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.

This is the first full-length book on the theme of symmetry in graphs, a fast-growing topic in algebraic graph theory. Suitable for graduate students, it goes from basic material on vertex-transitive graphs and permutation group theory right up to the field's major open problems, and includes many examples and exercises illustrating the theory.

This book is the first self-contained exposition of the link between dynamical systems and dimension groups. Recommended for graduate students as well as established researchers, it contains a thorough introduction to the field, including full proofs of the major results, as well a wealth of examples, exercises and open problems.

There has been a tremendous growth in understanding of random tilings over the past 25 years. This book, the first dedicated to the topic, caters to both beginning graduate students (or advanced undergraduates) wanting to learn the basics and to mature researchers looking to widen their background knowledge.

This book introduces the homological methods used in modern representation theory and discusses several landmark results that illustrate their power and beauty. With detailed exposition of many topics unavailable in one volume until now, it is an invaluable resource for advanced graduate students and researchers in representation theory.

Probabilistic number theory studies the many surprising interactions between whole numbers and the theory of random processes. This incisive textbook for beginning graduate students is the first to present and explain some of the most modern developments in the field, focusing on key examples and probabilistic ideas in the arguments.

Representation theory and character theory have proved essential in the study of finite simple groups since their early development by Frobenius. The author begins by presenting the foundations of character theory in a style accessible to advanced undergraduates requiring only a basic knowledge of group theory and general algebra.

This graduate textbook offers a self-contained introduction to the concepts and techniques of logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry and number theory. It features a systematic exposition of the foundations of the field, from the basic results on convex geometry and commutative monoids to the theory of logarithmic schemes and their de Rham and Betti cohomology. The book will be of use to graduate students and researchers working in algebraic, analytic, and arithmetic geometry as well as related fields.

Aimed at advanced students and active researchers in mathematics or theoretical physics, this book provides a detailed exposition of automorphic forms and representations, from the basics up to cutting-edge research topics at the interface between number theory and string theory.

This is Volume 1 of a two-volume book that provides a self-contained introduction to the analytical aspects of automorphic forms by proving several critical results carefully and in detail. With extensive examples, it will be useful for graduate students and researchers in automorphic forms, number theory, and other related fields.

This self-contained book introduces discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform. Featuring plenty of examples and over 160 exercises, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science.

The first exposition of group representations and harmonic analysis for graduates for over twenty years.

This two-volume text in harmonic analysis is appropriate for advanced undergraduate students with a strong background in mathematical analysis and for beginning graduate students wishing to specialize in analysis. With numerous exercises and problems it is suitable for independent study as well as for use as a course text.

This book is based on Professor Harper's experience in teaching global methods in combinatorial optimization and is ideal for graduate students as well as experienced researchers. The author has increased the utility of the text for teaching by including worked examples, exercises and material about applications to computer science.

This second edition of a classic work has been considerably expanded and revised, now with complete proofs of all results, including several new theorems not included in the first edition, such as Talagrand's generic chaining approach to boundedness of Gaussian processes and Gine and Zinn's characterization of uniform Donsker classes.

Based on the author's graduate course on association schemes and the optimal design of scientific experiments, this book is accessible to both pure mathematicians and statisticians alike. It will appeal to researchers as an accessible reference work from which to learn about the statistical/combinatorial aspect of their work.

Levy processes are perhaps the most basic class of stochastic processes with a continuous time parameter. This book provides the reader with comprehensive basic knowledge of Levy processes, and at the same time serves as an introduction to stochastic processes in general.

The many different tools from different fields that are used in additive combinatorics are brought together in a self-contained and systematic manner. This graduate-level 2006 text will quickly allow students and researchers easy entry into the fascinating field of additive combinatorics.

The purpose of this book is to study the relation between the representation ring of a finite group and its integral cohomology by means of characteristic classes. In this way it is possible to extend the known calculations and prove some general results for the integral cohomology ring of a group G of prime power order.

This book is intended to be used with graduate courses in Banach space theory.

This book provides probabilists with sufficient background to begin applying PDEs to probability theory and probability theory to PDEs. It covers the theory of linear and second order PDEs of parabolic and elliptic type. While most of the techniques described have antecedents in probability theory, the book does cover a few purely analytic techniques.

This 2003 book discusses the basic properties of period maps and period domains.

This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. Written in an informal style, this is a contemporary introduction to the subject. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.