This book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot's geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur's original treatment of the subject. In addition, each exercise is assigned a difficulty level to test readers' learning. Solutions and hints to most of the exercises are provided at the end.
This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral
This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and useful to graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral on Lipschitz curves
This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum–Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton–Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are
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.
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 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.
This gentle introduction to logic and model theory is based on a systematic use of three important games in logic: the semantic game; the Ehrenfeucht–Fraïssé game; and the model existence game. The third game has not been isolated in the literature before but it underlies the concepts of Beth tableaux and consistency properties. Jouko Väänänen shows that these games are closely related and in turn govern the three interrelated concepts of logic: truth, elementary equivalence and proof. All three methods are developed not only for first order logic but also for infinitary logic and generalized quantifiers. Along the way, the author also proves completeness theorems for many logics, including the cofinality quantifier logic of Shelah, a fully compact extension of first order logic. With over 500 exercises this book is ideal for graduate courses, covering the basic material as well as more advanced applications.
Two volume set of the authors' comprehensive and innovative work on multidimensional real analysis. These books are based on extensive teaching experience at Utrecht University and give a thorough account of analysis in multidimensional Euclidean space. They are an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects these books present an original treatment of the subject and they contain many results and exercises that cannot be found elsewhere. The numerous exercises illustrate a variety of applications in mathematics and physics. This combined with the exhaustive and transparent treatment of subject matter make these books ideal as either the text for a course, a source of problems for a seminar or for self study.
Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are
Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability, group representation theory, and the theory of integral operators. This unique text offers a unified overview of the topic, providing detailed examples of applications, as well as covering the fundamental underlying theory, including chapters on interpolation and approximation, Cholesky and Schur operations on kernels, and vector-valued spaces. Self-contained and accessibly written, with exercises at the end of each chapter, this unrivalled treatment of the topic serves as an ideal introduction for graduate students across mathematics, computer science, and engineering, as well as a useful reference for researchers working in functional analysis or its applications.
Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.
Lévy processes are rich mathematical objects and constitute perhaps the most basic class of stochastic processes with a continuous time parameter. This book is intended to provide the reader with comprehensive basic knowledge of Lévy processes, and at the same time serve as an introduction to stochastic processes in general. No specialist knowledge is assumed and proofs are given in detail. Systematic study is made of stable and semi-stable processes, and the author gives special emphasis to the correspondence between Lévy processes and infinitely divisible distributions. All serious students of random phenomena will find that this book has much to offer.
Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.
This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to estimate combinatorial quantities: generating functions are defined and their coefficients are then estimated via complex contour integrals. The multivariate case involves techniques well known in other areas of mathematics but not in combinatorics. Aimed at graduate students and researchers in enumerative combinatorics, the book contains all the necessary background, including a review of the uses of generating functions in combinatorial enumeration as well as chapters devoted to saddle point analysis, Groebner bases, Laurent series and amoebas, and a smattering of differential and algebraic topology. All software along with other ancillary material can be located via the book's website, http://www.cs.auckland.ac.nz/～mcw/Research/mvGF/asymultseq/ACSVbook/.
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.
