Bøger i Mathematical Surveys and Monographs serien

The third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.

Presents a self-contained account of applications of category theory to the theory of representations of algebras. The main focus is on 2-categorical techniques, including 2-categorical covering theory.

Gives a complete account, with full proofs, of the homotopy of $tmf$ and several $tmf$-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized.

Puts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results.

Provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory.

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.

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.

Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory.

Since its inception around 1980, the theory of perverse sheaves has been a vital tool of fundamental importance in geometric representation theory. This book gives a comprehensive account of constructible and perverse sheaves on complex algebraic varieties, and show how to put this to work in the context of geometric representation theory.

Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams.

Aims to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures. This book presents the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.

Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This two-volume monograph develops generalization of various topics in algebraic geometry in the context derived algebraic geometry.

The goal of this book is to present a portrait of the $n$-dimensional Cremona group with an emphasis on the 2-dimensional case. After recalling some crucial tools, the book describes a naturally defined infinite dimensional hyperbolic space on which the Cremona group acts.

The work on the foundations of Quantum Mechanics in the 1920s and 1930s provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic has developed in several directions. This book summarizes some of these directions.

Presents a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrodinger equation.

Explores applications of Jordan theory to the theory of Lie algebras. After presenting the general theory of nonassociative algebras and of Lie algebras, the book then explains how properties of the Jordan algebra attached to a Jordan element of a Lie algebra can be used to reveal properties of the Lie algebra itself.

Presents the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem.

Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behaviour of multiple ergodic averages. This book provides a comprehensive treatment of their role in ergodic theory.

This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach the completion of the proof of Theorem O.

Photonics and phononics encompass the fundamental science of light and sound propagation and interactions in complex structures, as well as its technological applications. This book reviews new mathematical tools, computational approaches, and inversion and optimal design methods to address challenging problems in photonics and phononics.

Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results.

Provides a road map to roughly 50 years of research detailing the role that the Fourier and Fourier-Stieltjes algebras have played in not only helping to better understand the nature of locally compact groups, but also in building bridges between abstract harmonic analysis, Banach algebras, and operator algebras. All of the important topics have been included.

Describes algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical $\mathcal{W}$-algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations.

The classification of finite simple groups is a landmark result of modern mathematics. This volume is part of a series of monographs being published by the AMS that represent the culmination of a century-long project. Part seven of the series, this is the middle volume of a trilogy treating the Generic Case.