Bøger i Mathematical Surveys and Monographs serien

Unitary representations of groups play an important role in many subjects, including number theory, geometry, probability theory, partial differential equations, and quantum mechanics. This monograph focuses on dual spaces associated to a group, which are spaces of building blocks of general unitary representations. Special attention is paid to discrete groups for which the unitary dual, the most common dual space, has proven to be not useful in general and for which other duals spaces have to be considered, such as the primitive dual, the normal quasi-dual, or spaces of characters. The book offers a detailed exposition of these alternative dual spaces and covers the basic facts about unitary representations and operator algebras needed for their study. Complete and elementary proofs are provided for most of the fundamental results that up to now have been accessible only in original papers and appear here for the first time in textbook form. A special feature of this monograph is that the theory is systematically illustrated by a family of examples of discrete groups for which the various dual spaces are discussed in great detail: infinite dihedral group, Heisenberg groups, affine groups of fields, solvable Baumslag-Solitar group, lamplighter group, and general and special linear groups. The book will appeal to graduate students who wish to learn the basics facts of an important topic and provides a useful resource for researchers from a variety of areas. The only prerequisites are a basic background in group theory, measure theory, and operator algebras.

Contains topics in Hamilton's Ricci flow. This book presents the theory of Ricci solitons, Kahler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology.

Presents some gratuitous generalities on scientific method as it relates to diffusion theory. This book defines Brownian motion by the characterization of P Levy, and then constructed in three basic ways and these are proved to be equivalent in the appropriate sense.

Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic $K$-theory and algebraic geometry. Suitable for graduate level, this title intends to obtain a deeper understanding of Quillen's model categories.

Around 1980, G Mason announced the classification of a certain subclass of a class of finite simple groups known as 'quasithin groups'. The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. This book offers a proof of a theorem classifying a larger class of groups.

Around 1980, G Mason announced the classification of a certain subclass of a class of finite simple groups known as 'quasithin groups'. The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. This book offers a proof of a theorem classifying a larger class of groups.

Over the years, the theory of Bergman spaces has undergone a remarkable metamorphosis. The subject of Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. This book is suitable for those who have studied basic complex function theory, measure theory, and functional analysis.

Presents a study of polynomial identities by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. This book includes such topics as polynomial rings in one or several variables, the Grassmann algebra, and finite-dimensional algebras. It is suitable for students of polynomial identity algebras.

Deals with the study of algebras of functions on quantum groups. This book includes the theory of Poisson Lie groups (quasi-classical version of algebras of functions on quantum groups), a description of representations of algebras of functions, and the theory of quantum Weyl groups.

Presents an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. This book contains several topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence.

Presents a geometric theory for incompressible flow and its applications to fluid dynamics. This monograph intends to study the stability and transitions of the structure of incompressible flows and its applications to fluid dynamics and geophysical fluid dynamics. It is suitable for researchers interested in nonlinear PDEs and fluid dynamics.

Studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. This book shows that Hopf algebras play a natural role in local Galois module theory.

Self-similar groups (groups generated by automata) initially appeared as examples of groups that are easy to define but have exotic properties like nontrivial torsion, intermediate growth, etc. This book studies the self-similarity phenomenon in group theory and shows its relationship with dynamical systems.

Treats free probability theory, which has been extensively developed since the early 1980s. This work focuses on entropy and the random matrix model approach. It is suitable as a reference for mathematicians interested in free probability theory and can serve as a text for an advanced graduate course.

Presents an outline of Alexander Grothendieck's theories. This book discusses four main themes - descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. It is suitable for those working in algebraic geometry.

Includes an extensive treatment of Hurwitz polynomials and other topics. This title features material on infrapolynomials, abstract polynomials, and matrix methods.

Riesz space (or a vector lattice) is an ordered vector space that is simultaneously a lattice. A topological Riesz space (also called a locally solid Riesz space) is a Riesz space equipped with a linear topology that has a base consisting of solid sets. This book presents a study (with complete proofs) of topological Riesz spaces.

Describes the structure of crossed products as revealed by the study of induced representations via the Green-Mackey-Rieffel machine. This book presents an analysis of the imprimitivity theorems on which Rieffel's theory of induced representations and Morita equivalence of $C^*$-algebras are based.

An exposition of what is known about the fundamental groups of compact Kahler manifolds. It collects together various results obtained over the years which aim to characterise those infinite groups which can arise as fundamental groups of compact Kahler manifolds. Most of these results are negative ones, saying which groups do not arise.

The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. This book provides a simple approach to basic hypergeometric series.

Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them.

The Teichmuller space $T(X)$ is the space of marked conformal structures on a given quasi conformal surface $X$. This book uses quasi conformal mapping to give a unified and up-to-date treatment of $T(X)$. It offers a treatment of deformations of complex structures on infinite Riemann surfaces.

The classification of finite simple groups is a landmark result of modern mathematics. This work presents critical aspects of the classification. It begins with the proof of a major theorem from the classification grid, namely Theorem $\mathcal{C}_7$. It is suitable for graduate students and researchers interested in group theory.

Presents a comprehensive exposition of the modern theory of valued and ordered fields. This title presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. It emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies them in a unified manner.

Gives a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. The authors have also attempted to give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

Summarizes the literature about the ring structure of the mod 2 cohomology of sporadic simple groups. This book offers background material on the relevant constructions from algebraic topology, and on local geometries from group theory.

Offering an introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, this book contains applications to arithmetic geometry and arithmetic dynamics. It presents a description of the topological structure of the Berkovich projective line and then introduces the Hsia kernel.

Presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization.