Bücher der Reihe Cambridge Studies in Advanced Mathematics

This corrected and clarified second edition, first published in 2006, includes a new chapter on the Riemannian geometry of surfaces and provides an introduction to the geometry of curved spaces. Its main theme is the effect of the curvature of spaces on the usual notions of geometry, angles, lengths, areas, and volumes, and on those new notions and ideas motivated by curvature itself.

L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with basic knowledge of classical analysis, complex variable theory, and algebra.

Two foremost researchers present important advances in stochastic process theory by linking well-understood (Gaussian) and less well-understood (Markov) classes of processes. It builds to this material through 'mini-courses' on the relevant ingredients, which assume only measure-theoretic probability. This original, readable 2006 book is for researchers and advanced graduate students.

This text provides the beginning graduate student, one with basic knowledge of real and functional analysis, with an account of p-summing and related operators. There are detailed expositions of the core results and highly non-trivial applications, for example to harmonic analysis, probability and measure theory and operator theory.

This book studies the geometric theory of polynomials and rational functions in the plane. Any theory in the plane should make full use of the complex numbers and thus the early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology and analysis.

To make the book accessible the authors have streamlined the presentation, referring the reader to the literature for purely technical results and working in a simplified setting where possible. In this way the new reader can penetrate rather deeply into the subject, whilst the book at the same time makes a useful reference for the more specialised reader.

This book provides a comprehensive account of the key theory on which the Taylor-Wiles proof of Fermat's last theorem is based, presenting an overview of the theory of automorphic forms on linear algebraic groups. The book will appeal to graduate students and researchers in number theory and arithmetic algebraic geometry.

This textbook on the calculus of variations covers from the basics to the modern aspects of the theory. It is intended to equip the reader to read present research papers in the subject.

This 1989 monograph investigates groups acting on low-dimensional topological spaces, and for the most part the viewpoint is algebraic. A familiarity with group theory is sufficient background for the first third of the book, while the later chapters occasionally state without proof and then apply various facts which require knowledge of homological algebra and algebraic topology.

Develops Galois theory in a much more general context than the undegraduate-style Galois theory of fields. In particular, the relationship with category theory is emphasized. Treats work by leading researchers in the field, and includes a comprehensive bibliography and index. Will be interesting for algebraists and category theorists.

This work provides the reader with a solid foundation in set theory, while the inclusion of topics such as absoluteness, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing, will usher the advanced reader to the doorstep of the research literature.

The techniques used to solve non-linear problems differ greatly from those dealing with linear features. Deriving all the necessary theorems from first principles, this 2005 textbook should give upper undergraduates and graduate students a thorough understanding using as little background material as possible.

The goal of this book is to lead the reader to an understanding of recent results on the Inverse Galois Problem. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory.

This volume is a self-contained introduction to the ideas needed to understand 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.

This 2001 introduction to Fourier analysis and partial differential equations is intended to be used with courses for beginning graduate students. With minimal prerequisites the authors take the reader from fundamentals to research topics in the area of nonlinear evolution equations, including the nonlinear Schroedinger and Kortweg-de Vries equations.

Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, to both the subject of the title and the Langlands correspondence for function fields. It is based on courses given by the author and will be welcomed by workers in number theory and representation theory.

The authors' aim in this book is to unite the seemingly disparate topics of Clifford algebras, analysis on manifolds and harmonic analysis. They show how algebra, geometry and differential equations all play a more fundamental role in Euclidean Fourier analysis.

A unique work giving a straightforward presentation of the logarithmic integral, a theme which lies athwart much of twentieth-century analysis.

An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory. This text is based on a graduate course given at Harvard University and is aimed at number theorists and algebraic geometers. Complex analysts and differential geometers will also find in it a clear account of recent results and applications of their subjects to new areas.

This work includes formal proof techniques, a section on applications of compactness, a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. At the same time, it exposes the foundations in an accessible manner, starting from informal calculations to give the novice a familiarity with the range of applications possible with spectral sequences.

This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra. The route chosen takes the reader quickly to the fundamental concepts for understanding complex projective geometry, the only prerequisites being a basic knowledge of linear and multilinear algebra and some elementary group theory.

Originally published in 1995, Cohomology of Drinfeld Modular Varieties provided an introduction, in two volumes, to both the subject of the title and the Langlands correspondence for function fields. It is based on courses given by the author and will be welcomed by workers in number theory and representation theory.

Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. An essential purchase for anyone carrying out research on wavelets and their applications.

Explicit Brauer Induction is an important technique in algebra, discovered by the author in 1986. It solves an old problem, giving a canonical formula for Brauer's induction theorem. In this 1994 book it is derived algebraically, following a method of R. Boltje - thereby making the technique, previously topological, accessible to algebraists.

This book gives a general outlook on homotopy theory. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given.

This 2003 volume consists of the first two volumes of Mukai's series on moduli theory. The work contained here is central to geometry but has also found applications in a wide range of other areas. This book will be a valuable resource for graduate students and researchers alike.

Metric Fixed Point Theory has proved a flourishing area of research for many mathematicians.

The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research.

Geometric control theory concerns the differential equations described by non-commuting vector fields. It builds ideas from the theory of differential systems and the calculus of variations into a cohesive mathematical framework applicable to a wide range of problems from differential geometry, applied mathematics, physics and engineering.