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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 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.

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.

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.

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.

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.

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.

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.

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green's algebras, the complete analysis of the random matchings, and the representation theory of the symmetric group.

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.

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.

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.

The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painleve transcendents, which have been termed the 'special functions of the twenty-first century'.

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 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.

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.

Originating with the pioneering works of P. Fatou and G. Julia, the subject of complex dynamics has seen great advances in recent years. Complex dynamical systems often exhibit rich, chaotic behavior, which yields attractive computer generated pictures, for example the Mandelbrot and Julia sets, which have done much to renew interest in the subject. This self-contained book discusses the major mathematical tools necessary for the study of complex dynamics at an advanced level. Complete proofs of some of the major tools are presented; some, such as the Bers-Royden theorem on holomorphic motions, appear for the very first time in book format. An appendix considers Riemann surfaces and Teichmuller theory. Detailing the very latest research, the book will appeal to graduate students and researchers working in dynamical systems and related fields. Carefully chosen exercises aid understanding and provide a glimpse of further developments in real and complex one-dimensional dynamics.

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 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 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.

The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.

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.

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.

Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

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 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.

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 representation theory of the symmetric groups is a classical topic that, since the pioneering work of Frobenius, Schur and Young, has grown into a huge body of theory, with many important connections to other areas of mathematics and physics. This self-contained book provides a detailed introduction to the subject, covering classical topics such as the Littlewood-Richardson rule and the Schur-Weyl duality. Importantly the authors also present many recent advances in the area, including Lassalle's character formulas, the theory of partition algebras, and an exhaustive exposition of the approach developed by A. M. Vershik and A. Okounkov. A wealth of examples and exercises makes this an ideal textbook for graduate students. It will also serve as a useful reference for more experienced researchers across a range of areas, including algebra, computer science, statistical mechanics and theoretical physics.

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.