Representation Theory of Finite Groups: Algebra and Arithmetic (Graduate Studies in Mathematics, 59)

This is the first comprehensive introduction to the theory of mass transportation with its many--and sometimes unexpected--applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of "optimal transportation" (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. The Book Assumes Only The Material Of A Standard Graduate Course In Algebra. It Is Suitable As A Text For A Year-long Graduate Course. The Subject Is Of Interest To Students Of Algebra, Number Theory And Algebraic Geometry. The Systematic Treatment Presented Here Makes The Book Also Valuable As A Reference.--book Jacket. Ch. 1. Introduction -- Ch. 2. Semisimple Rings And Modules -- 2.1. Basic Notions -- 2.2. Structure Theorems -- 2.3. Idempotents And Blocks -- 2.4. Behavior Under Field Extensions -- 2.5. Theorems Of Burnside And Frobenius-schur -- Ch. 3. Semisimple Group Representations -- 3.1. Examples And General Results -- 3.2. Representations Of Abelian Groups -- 3.3. Decomposition Of The Regular Representation -- 3.4. Applications Of Frobenius's Theorem -- 3.5. Characters -- 3.6. Idempotents And Their Uses -- 3.7. Subfields Of The Complex Numbers -- 3.8. Fields Of Positive Characteristic -- Ch. 4. Induced Representations And Applications -- 4.1. Induced Representations -- 4.2. Mackey's Theorem -- 4.3. Permutation Representations -- 4.4. M-groups -- 4.5. Theorems Of Artin And Brauer -- 4.6. Degrees Of Irreducible Representations -- Ch. 5. Introduction To Modular Representations -- Ch. 6. General Rings And Modules -- 6.1. Jordan-holder And Krull-schmidt Theorems -- 6.2. The Jacobson Radical -- 6.3. Rings Of Finite Length -- 6.4. Finite-dimensional Algebras -- Ch. 7. Modular Group Representations -- 7.1. General Results -- 7.2. Characters And Brauer Characters -- 7.3. Examples -- App. Some Useful Results. Steven H. Weintraub. Includes Bibliographical References (p. 209) And Index. Cedric Villani's book is a lucid and very readable documentation of the tremendous recent analytic progress in “optimal mass transportation” theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics. —Lawrence C. Evans, University of California at Berkeley In 1781, Gaspard Monge defined the problem of “optimal transportation”, or the transferring of mass with the least possible amount of work, with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is at once an introduction to the field of optimal transportation and a survey of the research on the topic over the last 15 years. The book is intended for graduate students and researchers, and it covers both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference. Ch. 1. The Kantorovich Duality -- Ch. 2. Geometry Of Optimal Transportation -- Ch. 3. Brenier's Polar Factorization Theorem -- Ch. 4. The Monge-ampere Equation -- Ch. 5. Displacement Interpolation And Displacement Convexity -- Ch. 6. Geometric And Gaussian Inequalities -- Ch. 7. The Metric Side Of Optimal Transportation -- Ch. 8. A Differential Point Of View On Optimal Transportation -- Ch. 9. Entropy Production And Transportation Inequalities -- Ch. 10. Problems -- Table Of Short Statements. Cédric Villani. Includes Bibliographical References (p. 349-362) And Index. This graduate textbook reviews results about the existence and regularity of the optimal mass transportation problem, introducing such topics as displacement interpolation, its applications to functional inequalities with geometric content, the Monge-Kantorovich problem, and a differential formulation of the transportation problem inspired by fluid mechanics. Annotation (c)2003 Book News, Inc., Portland, OR A graduate textbook introducing group representation theory to students who have no prior knowledge of it, and may not even know what a representation is. They are expected to have a sound knowledge of linear algebra and a good familiarity with basic module theory. Annotation (c) Book News, Inc., Portland, OR (booknews.com) Presents a comprehensive introduction to the theory of mass transportation with its many - and sometimes unexpected - applications. This book defines the problem of 'optimal transportation' (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. Assume that we are given a pile of sand (say), and a hole that we have to completely fill up with the sand. We shall begin by giving some basic examples of the objects and questions we will be studying.