Algebra Books

Book SynopsisThis volume is the proceedings of the 5th MSJ Seasonal Institute 'Schubert Calculus' held at Osaka City University, September 17th-27th, 2012. It is recommended for all researchers and graduate students who are interested in Schubert calculus and its many connections and applications to related areas of mathematics, such as geometric representation theory, combinatorial aspects of algebraic varieties arising in Lie theory, and equivariant topology. Alain Lascoux, who is one of the pioneers of modern Schubert calculus, and a contributor of this volume, passed away during the time of editing process of the proceedings. The volume is dedicated to him.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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Book SynopsisThe discovery of the algorithm by Bruno Buchberger on July 1965, so-called the Buchberger algorithm, to compute Gröbner bases of ideals of the polynomial ring had led the birth of the exciting research area called 'Computer Algebra' in the modern mathematics. The 8th Mathematical Society of Japan Seasonal Institute (MSJ SI 2015) entitled 'The 50th Anniversary of Gröbner Bases' was held on July 2015, which is the 50th year following the discovery of the Buchberger algorithm. This volume is the proceedings of MSJ SI 2015 and consists of 14 papers related with computer algebra, algebraic statistics, D-modules, convex polytopes and toric ideals. These papers enable the reader to look over the current trends on Gröbner bases. Especially, young researchers can reach a treasury of fascinating research problems which are pending. Foreword was contributed by Bruno Buchberger, where a secret story on the discovery of the Buchberger algorithm is stated.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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Book SynopsisThis volume is the proceedings of the online conference 'McKay correspondence, Mutation and Related Topics,' which was held at Kavli Institute for the Physics and Mathematics of the Universe (IPMU) in Japan in Summer in 2020.This volume consists of 15 peer-reviewed articles written by speakers who lead currently active research fields of McKay correspondence, crepant resolution, singularities, quiver and cluster algebra.The editors believe that this volume is quite useful for all researchers and students who are interested in these fruitful research fields.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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Book SynopsisThis book provides an introduction to the algebraic theory of semirings and, in this context, to basic algebraic concepts as e.g. semigroups, lattices and rings. It includes an algebraic theory of infinite sums as well as a detailed treatment of several applications in theoretical computer science. Complete proofs, various examples and exercises (some of them with solutions) make the book suitable for self-study. On the other hand, a more experienced reader who looks for information about the most common concepts and results on semirings will find cross-references throughout the book, a comprehensive bibliography and various hints to it.Table of ContentsBasic concepts; extensions of semirings; partially-ordered semirings; semirings with infinite sums; semialgebras; semigroup semirings and semirings of formal power series.

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Book SynopsisThis book is an introduction to the theory and application of modes — structures that capture the common underlying algebra of convex sets, affine spaces and certain ordered sets. Modes appear in many branches of mathematics, particularly geometry and combinatorics, and have been used in computer science, economics, physics, and biology. The initial stage of the theory was set out in the authors' research monograph Modal Theory (published in 1985). The present book provides a more complete theory, the result of research conducted during the subsequent 15 years. It contains a clear introduction to selected topics from universal algebra, category theory and model theory, and the foundations of the theory of modes, as well as more advanced topics leading to the forefront of current research in the field.The authors have included a wide range of exercises, usually placed at the end of the section, and indexed alphabetically. Some exercises are simply designed to familiarize readers with the notation and concepts. Others are more difficult, extending the content of the sections in which they appear, and providing a foretaste of further research.Table of ContentsAlgebras; Categories of Algebras; Varieties, Prevarieties and Quasivarieties; Constructing Algebras and Quasivarieties; Introduction to Modes; Mal'cev Modes and Affine Spaces; Subreducts of Affine Spaces; Binary Modes; Hierarchical Information Theory; Recent Developments and Open Problems.

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Book SynopsisOver the past 20 years, the theory of groups — in particular simple groups, finite and algebraic — has influenced a number of diverse areas of mathematics. Such areas include topics where groups have been traditionally applied, such as algebraic combinatorics, finite geometries, Galois theory and permutation groups, as well as several more recent developments. Among the latter are probabilistic and computational group theory, the theory of algebraic groups over number fields, and model theory, in each of which there has been a major recent impetus provided by simple group theory. In addition, there is still great interest in local analysis in finite groups, with substantial new input from methods of geometry and amalgams, and particular emphasis on the revision project for the classification of finite simple groups.This important book contains 20 survey articles covering many of the above developments. It should prove invaluable for those working in the theory of groups and its applications.Table of ContentsCurtis-Phan-Tits Theory, C.D. Bennett et al; Derangements in Simple and Primitive Groups, J. Fulman & R. Guralnick; Computing with Matrix Groups, W.M. Kantor & A. Seress; Bases of Primitive Permutation Groups, M.W. Liebeck & A. Shalev; Modular Subgroup Arithmetic, T.W. Muller; Counting Nets in the Monster, S.P. Norton; Overgroups of Finite Quasiprimitive Permutation Groups, C.E. Praeger; Old Groups Can Learn New Tricks, L. Pyber; Structure and Presentations of Lie-Type Groups, F.G. Timmesfeld; Computing in the Monster, R.A. Wilson; and other papers.

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Book SynopsisThis short but rigorous book approaches the main ideas of linear algebra through carefully selected examples and relevant applications. It is intended for students with various interests in mathematics, as well as established scientists seeking to refresh their basic mathematical culture.The book is also a good introduction to functional analysis and quantum theory since it discusses the general principles of linear algebra without finiteness assumptions.Table of Contents# Linear Systems: Elimination Method # Vector Spaces # Matrix Multiplication # Linear Maps # The Rank Theorem # Eigenvectors and Eigenvalues # Inner-Product Spaces # Symmetric Operators # Duality # Determinants # Applications # Normal Operators

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Book SynopsisThe area of automorphic representations is a natural continuation of studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called “liftings”. This book concentrates on two initial examples: the symmetric square lifting from SL(2) to PGL(3), reflecting the 3-dimensional representation of PGL(2) in SL(3); and basechange from the unitary group U(3, E/F) to GL(3, E), [E : F] = 2.The book develops the technique of comparison of twisted and stabilized trace formulae and considers the “Fundamental Lemma” on orbital integrals of spherical functions. Comparison of trace formulae is simplified using “regular” functions and the “lifting” is stated and proved by means of character relations.This permits an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), a proof of multiplicity one theorem and rigidity theorem for SL(2) and for U(3), a determination of the self-contragredient representations of PGL(3) and those on GL(3, E) fixed by transpose-inverse-bar. In particular, the multiplicity one theorem is new and recent.There are applications to construction of Galois representations by explicit decomposition of the cohomology of Shimura varieties of U(3) using Deligne's (proven) conjecture on the fixed point formula.Table of ContentsOn the Symmetric Square Lifting: Functoriality and Norms; Orbital Integrals; Twisted Trace Formula; Total Global Comparison; Applications of a Trace Formula; Computation of a Twisted Character; Automorphic Representations of the Unitary Group U(3, E/F): Local Theory; Trace Formula; Liftings and Packets; Zeta Functions of Shimura Varieties of U(3): Automorphic Representations; Local Terms; Real Representations; Galois Representations.

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Book SynopsisThis volume provides a comprehensive treatment of basic Lie theory, primarily directed toward graduate study. The text is ideal for a full graduate course in Lie groups and Lie algebras. However, the book is also very usable for a variety of other courses: a one-semester course in Lie algebras, or on Haar measure and its applications, for advanced undergraduates; or as the text for one-semester graduate courses in Lie groups and symmetric spaces of non-compact type, or on lattices in Lie groups. The material is complete and detailed enough to be used for self-study; it can also serve as a reference work for professional mathematicians working in other areas. The book's utility for such a varied readership is enhanced by a diagram showing the interdependence of the separate chapters so that individual chapters and the material they depend upon can be selected, while others can be skipped.The book incorporates many of the most significant discoveries and pioneering contributions of the masters of the subject: Borel, Cartan, Chevalley, Iwasawa, Mostow, Siegel, and Weyl, among others.Table of ContentsLie Groups and Lie Algebras: Introduction; Lie Groups; Haar Measure and its Applications; Elements of the Theory of Lie Algebras; The Structure of Compact Connected Lie Groups; Representations of Compact Lie Groups; Symmetric Spaces of Non-Compact Type; Semisimple Lie Algebras and Lie Groups; Lattices in Lie Groups; Density Results for Co-Finite Volume Subgroups.

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Book SynopsisThis accessible book for beginners uses intuitive geometric concepts to create abstract algebraic theory with a special emphasis on geometric characterizations. The book applies known results to describe various geometries and their invariants, and presents problems concerned with linear algebra, such as in real and complex analysis, differential equations, differentiable manifolds, differential geometry, Markov chains and transformation groups. The clear and inductive approach makes this book unique among existing books on linear algebra both in presentation and in content.Table of ContentsThe Euclidean Structures of R1, R2 and R3: The Euclidean Plane R2; The Euclidean Space R3.

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Book SynopsisThis textbook comprehensively introduces students and researchers to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. Covering all the modern techniques in detail, it relates applications to cutting-edge research fields such as Yang-Mills theory and string theory.Aimed at readers in applied mathematics and physics rather than pure mathematics, the material is ideally suited to students and researchers whose main interest lies in finding solutions to differential equations and invariants of maps.A large number of worked examples and challenging exercises help readers to work independently of teachers, and by including SymbolicC++ implementations of the techniques in each chapter, the book takes full advantage of the advancements in algebraic computation.Twelve new sections have been added in this edition, including: Haar measure, Sato's theory and sigma functions, universal algebra, anti-self dual Yang-Mills equation, and discrete Painlevé equations.Table of ContentsGroups; Lie Groups and Lie Transformation Groups; Infinitesimal Transformations; Lie Algebras; Introductory Examples; Differential Forms and Tensor Fields; Lie Derivative and Invariance; Invariance of Differential Equations; Lie-Backlund Vector Fields; Differential Equation for a Given Lie Algebra; A List of Lie Symmetry Vector Fields; Recursion Operators; Backlund Transformations; Lax Representations; Conservation Laws; Symmetries and Painleve Test; Lie Algebra Valued Differential Forms; Bose Operators and Lie Algebras; Maps and Invariants; Computer Algebra; Differential Manifolds.

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Book SynopsisRing theory has been developing through the interaction between the investigation of its own algebraic structure and its application to many areas of mathematics, computer science, and physics among others.This volume consists of a collection of survey articles by invited speakers and original articles refereed by world experts that was presented at the fifth China-Japan-Korea International Symposium. The survey articles provide some ideas of the application as well as an excellent overview of the various areas in ring theory. The original articles exhibit new ideas, tools and techniques needed for successful research investigation in ring theory and show the trend of current research.The articles cover all of the most important areas in ring theory, making this volume a useful resource book for researchers in mathematics — both beginners and advanced experts.Table of ContentsQuasi-Baer Rings; Endomorphism Rings; Torsion-Free Modules; Graded Algebras; Finitistic Dimensions; Projective Modules; ArtinA-Schelter Regular Algebras; Noncommutative Valuation Rings; Noncommutative Algebraic Geometry; AuslanderA-Gorenstein Rings; QF Rings; Hochschild Cohomology; Jordan Derivations in Rings; Harada Rings; Quaternoin Algebras; Representation Theory; Rings of Quotients; Lie Algebras; Skew Polynomial Rings; Hopf Algebras.

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Book SynopsisThe Shape of Algebra in the Mirrors of Mathematics is a unique text aiming to explain some elements of modern mathematics and to show its flavor and unity. It is neither a standard textbook nor a tour of algebra for a casual reader. Rather, it is an attempt to share authors' mathematical experiences and philosophy with readers who have more than a passing interest in mathematics, but only a traditional exposure to High School Algebra and some elements of Calculus. The book gives the readers a sense of visual nature of significant part of mathematics.The reader becomes an owner of and a researcher in VisuMatica, a virtual math laboratory. It is an original and comprehensive PC software package (a brainchild of the second author) that will enable the reader to experience mathematics both as a human intellectual endeavor and as an experimental science. Although it is possible to read and appreciate the book without ever visiting the VisuMatica lab, those who engage with the interactive demos found in the lab will greatly advance their understanding of the text. The book seeks to encourage an interactive, investigative style of learning that can promote the habits of mind characteristic of modern mathematical thinking.An outline of the topics that are discussed may read like a list of graduate courses: Abstract Algebra, Topology, Singularity Theory, Complex Analysis, and Number Theory. However, they are presented from an intuitive perspective that uses primarily visual models and concepts. Although the main subject is polynomials and polynomial equations, the true story line is the interplay between basic ideas from algebra, geometry, analysis and topology.The Shape of Algebra might serve as a text for an “appreciation” course in modern mathematics designed for non-mathematics majors or for students who are considering majoring in mathematics or related disciplines. The authors' goal is to present the reader with a fresh viewpoint on the sense and flavor of mathematics. The subject is often presented in a fashion that students find stale and uncompelling. The book's emphasis, in contrast, is on how a modern practitioner thinks about and works within the discipline.The book aims to attract students of all ages, particularly including professional mathematicians interested in pedagogy. In part, The Shape of Algebra is directed at secondary and college teachers and students who want to expand their horizons in the field and to find both a fresh presentation of familiar concepts and, perhaps, some unexpected results. Many of the topics and demos are self-contained and can be used individually to enhance traditional courses. Several of the ideas and materials developed in the book have been tested in high school and college classrooms. The book will enable readers to approach its content on three levels: the first level requires only some fluency with routine algebraic manipulations; the second also presumes familiarity with the notions of derivatives, and the third uses some basic concepts of multivariable calculus and linear algebra. All three levels are clearly marked in the text, and allow for a smooth reading enhanced by virtual experiments.Table of ContentsIntroduction - Maps upon Maps; Maps, Models, and the Coordinate Plane; The Universe of Quadratic Polynomials; The Complex Numbers; The Geometry of Complex Linear Polynomial Maps; Complex Polynomial Maps, Closed Plane Curves and their Invariants; Experimenting with the Geometry of Complex Polynomial Maps of Degrees Two and Three; The Old Art of Solving Cubic Equations and the New Science; Complex Polynomial Maps of Higher Degrees and the Fundamental Theorem of Algebra; Local and Global Geometry of Complex Polynomial Maps; The Fundamental Theorem and the Complex Viete Map; Appendices: Estimating Polynomial Expressions; More about the Taylor Formula for Polynomials; The Geometry of Discriminants; Line Integrals and an Analytical Proof of the Loop Theorem; Deformations of Complex Polynomials and the Braids.

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Book SynopsisThis book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type when applied to certain residual Eisenstein series. This book contains a complete account of this automorphic descent, with complete, detailed proofs. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.Table of ContentsCertain Residual Eisenstein Series; Fourier Coefficients of Gelfand-Graev Type and Fourier-Jacobi Type; Jacquet Modules Corresponding to Gelfand-Graev Models; Jacquet Modules Corresponding to Fourier-Jacobi Models; The Tower Property; Non-Vanishing of the Descent; On the Global Genericity of the Descent and Rankin-Selberg Integrals; The Descent and Langlands Functoriality from Classical Groups to GL(n); Applications of the Descent (Generalized Endoscopy, Base Change).

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Book SynopsisAddresses various aspects of the theory of automorphic forms and its relations with the theory of $L$-functions, the theory of elliptic curves, and representation theory. This volume is intended for researchers interested in expanding their own areas of focus, thus allowing them to build bridges to mathematical questions in other fields.Table of Contents S. Anni, A note on the minimal level of realization for a mod $\ell$ eigenvalue system A. Arnold-Roksandich, A discussion on the number eta-quotients of prime level C. Burrin, Dedekind sums, reciprocity, and non-arithmetic groups G. Chinta, I. Horozov, and C. O'Sullivan, Noncommutative modular symbols and Eisenstein series A. Espinosa, An annotated discussion of a panel presentation on improving diversity in mathematics J. S. Friedman, J. Jorgenson, and L. Smajlovic, Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps X. Guitart and M. Masdeu, Computing $p$-adic periods of abelian varieties from automorphic forms A. Haensch and B. Kane, An algebraic and analytic approach to spinor exceptional behavior in translated lattices A. K. Jha and B. Sahu, Differential operators on Jacobi forms and special values of certain Dirichlet series J. Jorgenson and L. Smajlovic, Some results in study of Kronecker limit formula and Dedekind sums D. Kelmer, Equidistribution of shears and their arithmetic applications K. Khuri-Makdisi, Fake proofs for identities involving products of Eisenstein series K. Khuri-Makdisi, Modular forms constructed from moduli of elliptic curves, with applications to explicit models of modular curves B. Kumar, J. Meher, and S. Pujahari, Some remarks on the coefficients of symmetric power $L$-functions J. Li, On primes in arithmetic progressions B. Linowitz and L. Thompson, The Fourier coefficients of Eisenstein series newforms K. Maurischat, Properties of Sturm's formula A. Odzak and L. Sceta, An application of a special form of a Tauberian theorem A. Odzak and L. Sceta, On the zeros of some $L$ functions from the extended Selberg class E. Ozman, Rational points on twisted modular curves B. Ramakrishnan, B. Sahu, and A. K. Singh, On the number of representations of certain quadratic forms in 8 variables M. Roy, Level of Siegel modular forms constructed via $\textrm{sym}^3$ lifting F. Stromberg, Dimension formulas and kernel functions for Hilbert modular forms H. Then, An explicit evaluation of the Hauptmoduli at elliptic points for certain arithmetic groups A. Trbovic, Torsion groups of elliptic curves over quadratic fields S. Wagh, Maass space for lifting from SL(2,$\mathbb{R}$) to GL(2,B) over a division quaternion algebra N. Walji, On the occurrence of large positive Hecke eigenvalues for GL(2) L. H. Walling, Representations by quadratic forms and the Eichler Commutation Relation S. Yamana, Degenerate principal series and Langlands classification.

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Book Synopsis

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Book Synopsis

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Book SynopsisThe objective of this book is to look at certain commutative graded algebras that appear frequently in algebraic geometry. By studying classical constructions from geometry from the point of view of modern commutative algebra, this carefully-written book is a valuable source of information, offering a careful algebraic systematization and treatment of the problems at hand, and contributing to the study of the original geometric questions. In greater detail, the material covers aspects of rational maps (graph, degree, birationality, specialization, combinatorics), Cremona transformations, polar maps, Gauss maps, the geometry of Fitting ideals, tangent varieties, joins and secants, Aluffi algebras. The book includes sections of exercises to help put in practice the theoretic material instead of the mere complementary additions to the theory.

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Book SynopsisOver a career that spanned 60 years, Ronald L. Graham (known to all as Ron) made significant contributions to the fields of discrete mathematics, number theory, Ramsey theory, computational geometry, juggling and magical mathematics, and many more. Ron also was a mentor to generations of mathematicians, he gave countless talks and helped bring mathematics to a wider audience, and he held signifi cant leadership roles in the mathematical community. This volume is dedicated to the life and memory of Ron Graham, and includes 20-articles by leading scientists across a broad range of subjects that refl ect some of the many areas in which Ron worked.

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Book SynopsisThis comprehensive reference begins with a review of the basics followed by a presentation of flag varieties and finite- and infinite-dimensional representations in classical types and subvarieties of flag varieties and their singularities. Associated varieties and characteristic cycles are covered as well and Kazhdan-Lusztig polynomials are treated. The coverage concludes with a discussion of pattern avoidance and singularities and some recent results on Springer fibers.

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Book SynopsisContinuous model theory is an extension of classical first order logic which is best suited for classes of structures which are endowed with a metric. Applications have grown considerably in the past decade. This book is dedicated to showing how the techniques of continuous model theory are used to study C*-algebras and von Neumann algebras. This book geared to researchers in both logic and functional analysis provides the first self-contained collection of articles surveying the many applications of continuous logic to operator algebras that have been obtained in the last 15 years.

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Book SynopsisThis proceedings volume documents the contributions presented at the CONIAPS XXVII International Conference on Recent Advances in Pure and Applied Algebra. The entries focus on modern trends and techniques in various branches of pure and applied Algebra and highlight their applications in coding, cryptography, graph, and fuzzy theory. The book comprised a total of eighteen chapters, among which the first fourteen chapters are devoted to Algebra and related topics, and the last four chapters are included applied mathematics parts. The chapters present the latest research work being done on the frontiers of the various branches of algebra as well as showcase the cross-fertilization of the ideas and connection among these branches.Covering a broad range of topics in pure and applied Algebra, this volume would appeal to a wide spectrum of the researcher in Mathematics. The main aim of this monograph is to contribute to the development of pure and applied Algebra and hence we purposely sought a cross-section of topics in Algebra and encouraged expository presentations and research papers that provide an innovative link between research areas of Algebra and the field of their applications. This volume will be useful not only to experts but also to beginners of research in algebras and related topics.

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Book SynopsisNoncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.

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Book SynopsisThis is the first in a series of three volumes dealing with important topics in algebra. It offers an introduction to the foundations of mathematics together with the fundamental algebraic structures, namely groups, rings, fields, and arithmetic. Intended as a text for undergraduate and graduate students of mathematics, it discusses all major topics in algebra with numerous motivating illustrations and exercises to enable readers to acquire a good understanding of the basic algebraic structures, which they can then use to find the exact or the most realistic solutions to their problems.Trade Review“The rich material is explaind in a very lucid, detailed and rigorous style. Together with the unique wealth of instructive illustrations, examples, and exercises, the main text makes an excellent source for both teaching and self-study of the subject. Certainly, this new textbook is a very useful and valuable enrichment of the plenty of existing primers in the field of higher algebra.” (Werner Kleinert, zbMATH 1369.00002, 2017)Table of ContentsChapter 1. Language of mathematics 1 (Logic).- Chapter 2. Language Of Mathematics 2 (Set Theory).- Chapter 3. Number System.- Chapter 4. Group Theory.- Chapter 5. Fundamental Theorems.- Chapter 6. Permutation groups and Classical Groups.- Chapter 7. Elementary Theory of Rings and Fields.- Chapter 8. Number Theory 2.- Chapter 9. Structure theory of groups.- Chapter 10. Structure theory continued.- Chapter 11. Arithmetic in Rings.

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