The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

In this well-illustrated book the authors, Sinan Kanbir, Ken Clements, and Nerida Ellerton, tackle a persistent, and universal, problem in school mathematics-why do so many middle-school and secondary-school students find it difficult to learn algebra well? What makes the book important are the unique features which comprise the design-research approach that the authors adopted in seeking a solution to the problem. The first unique feature is that the authors offer an overview of the history of school algebra. Despite the fact that algebra has been an important component of secondary-school mathematics for more than three centuries, there has never been a comprehensive historical analysis of factors influencing the teaching and learning of that component. The authors identify, through historical analysis, six purposes of school algebra: (a) algebra as a body of knowledge essential to higher mathematical and scientific studies, (b) algebra as generalized arithmetic, (c) algebra as a prerequisite for entry to higher studies, (d) algebra as offering a language and set of procedures for modeling real-life problems, (e) algebra as an aid to describing structural properties in elementary mathematics, and (f) algebra as a study of variables. They also raise the question whether school algebra represents a unidimensional trait. Kanbir, Clements and Ellerton offer an unusual hybrid theoretical framework for their intervention study (by which seventh-grade students significantly improved their elementary algebra knowledge and skills). Their theoretical frame combined Charles Sanders Peirce's triadic signifier-interpretant-signified theory, which is in the realm of semiotics, with Johann Friedrich Herbart's theory of apperception, and Ken Clements' and Gina Del Campo's theory relating to the need to expand modes of communications in mathematics classrooms so that students engage in receptive and expressive modes. Practicing classroom teachers formed part of the research team. This book appears in Springer's series on the "History of Mathematics Education." Not only does it include an important analysis of the history of school algebra, but it also adopts a theoretical frame which relies more on "theories from the past," than on contemporary theories in the field of mathematics education. The results of the well-designed classroom intervention are sufficiently impressive that the study might havecreated and illuminated a pathway for future researchers to take.

This book explores the role of listening in community engagement and peace building efforts, bridging academic research in communication and practical applications for individual and social change. For all their differences, community engagement and peacebuilding efforts share much in common: the need to establish and agree on achievable and measurable goals, the importance of trust, and the need for conflict management, to name but a few. This book presents listening-considered as a multi-disciplinary concept related to but distinct from civility, civic participation, and other social processes-as a primary mechanism for accomplishing these tasks. Individual chapters explore these themes in an array of international contexts, examining topics such as conflict resolution, restorative justice, environmental justice, migrants and refugees, and trauma-informed peacebuilding. The book includes contemporary literature reviews and theoretical insights covering the role of listening as related to individual, social, and governmental efforts to better engage communities and build, maintain, or establish peace in an increasingly divided world. This collection provides invaluable insight to researchers, students, educators, and practitioners in intercultural and international communication, conflict management, peacebuilding, community engagement, and international studies.

Evolution and the Human-Animal Drive to Conflict examines how fundamental, universal animal drives, such as dominance/prevalence, survival, kinship, and "profit" (greed, advantage, whether of material or social nature), provide the basis for 'the evolutionary trap' that promotes the unstable, conflictive, dominant-prone individual and group human behaviours. Examining this behavioural tension, the book argues that while these innate features set up behaviours that lean towards aggression influenced by social inequalities, the means implemented to defuse them resort to emotional and intellectual strategies that sponsor fanaticism and often reproduce the very same behaviours they intend to defuse. In addressing these concerns, the book argues that we should enhance our resources to promote solidarity, accept cultural differences, deter expansionist and uncontrolled profit drives, and achieve collective access towards knowledge and progress in living conditions. This entails promoting the redistribution of resources and creative labour access and avoiding policies that generate a fragmented world with collective and individual development disparities that invite and encourage dominance behaviours.This resource redistribution asserts that it is necessary to reformulate the global set of human priorities towards increased access to better living conditions, cognitive enhancement, a more amiable interaction with the ecosystem and non-aggressive cultural differences, promote universal access to knowledge and enhance creativity and cultural convivence. These behavioural changes entail partial derangement of our ancestral animal drives camouflaged under different cultural profiles until the species succeeds in replacing the dominance of basic animal drives with prosocial, collective ones. Though it entails a formidable task of confronting financial, military, and religious powers and cultural inertias - human history is also a challenging, continuous experience in these domains - for the sake of our own self-identity and self-evaluation we should reject any suggestion of not continuing embracing slowly constructing collective utopias channelled towards improving individual and collective freedom and creativeness. This book will interest academics and students in social, cognitive, and evolutionary psychology, the neurosciences, paleoanthropology, philosophy, and anthropology.

The main purpose of these lectures is first to briefly survey the fundamental con nection between the representation theory of the symmetric group Sn and the theory of symmetric functions and second to show how combinatorial methods that arise naturally in the theory of symmetric functions lead to efficient algorithms to express various prod ucts of representations of Sn in terms of sums of irreducible representations. That is, there is a basic isometry which maps the center of the group algebra of Sn, Z(Sn), to the space of homogeneous symmetric functions of degree n, An. This basic isometry is known as the Frobenius map, F. The Frobenius map allows us to reduce calculations involving characters of the symmetric group to calculations involving Schur functions. Now there is a very rich and beautiful theory of the combinatorics of symmetric functions that has been developed in recent years. The combinatorics of symmetric functions, then leads to a number of very efficient algorithms for expanding various products of Schur functions into a sum of Schur functions. Such expansions of products of Schur functions correspond via the Frobenius map to decomposing various products of irreducible representations of Sn into their irreducible components. In addition, the Schur functions are also the characters of the irreducible polynomial representations of the general linear group over the complex numbers GLn(C)."

Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects. The central notion in the book is that of expansion, which roughly means the quality of a graph as a communications network. Cayley graphs are certain graphs constructed from groups; they play a prominent role in the study of expander families. The isoperimetric constant, the second largest eigenvalue, the diameter, and the Kazhdan constant are four measures of the expansion quality of a Cayley graph. The book carefully develops these concepts, discussing their relationships to one another and to subgroups and quotients as well as their best-case growth rates. Topics include graph spectra (i.e., eigenvalues); a Cheeger-Buser-type inequality for regular graphs; group quotients and graph coverings; subgroups and Schreier generators; the Alon-Boppana theorem on the second largest eigenvalue of a regular graph; Ramanujan graphs; diameter estimates for Cayley graphs; the zig-zag product and its relation to semidirect products of groups; eigenvalues of Cayley graphs; Paley graphs; and Kazhdan constants. The book was written with undergraduate math majors in mind; indeed, several dozen of them field-tested it. The prerequisites are minimal: one course in linear algebra, and one course in group theory. No background in graph theory or representation theory is assumed; the book develops from scatch the required facts from these fields. The authors include not only overviews and quick capsule summaries of key concepts, but also details of potentially confusing lines of reasoning. The book contains ideas for student research projects (for capstone projects, REUs, etc.), exercises (both easy and hard), and extensive notes with references to the literature.

This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.

This volume contains the proceedings of the conference A Panorama on Singular Varieties, celebrating the 70th birthday of Le Dung Trang, held from February 7-10, 2017, at the University of Seville, IMUS, Seville, Spain. The articles cover a wide range of topics in the study of singularities and should be of great value to graduate students and research faculty who have a basic background in the theory of singularities. This book is published in cooperation with Real Sociedad Matematica Espanola.

Insecurity is an inevitable part of being human. Although life is insecure for every organism, humans alone are burdened by knowing that this is so. This ground-breaking volume features contributions by leading international researchers exploring the social psychology of insecurity, and how existential, metaphysical and social uncertainty influence human social behaviour. Chapters in the book investigate the psychological origins of insecurity, evolutionary theorizing about the functions of insecurity, the motivational strategies people adopt to manage insecurity, self-regulation strategies, the role of insecurity in the formation and maintenance of social relationships, and the influence of insecurity and uncertainty on the organization of larger social systems and public affairs. The chapters also discuss how insecurity influences many areas of contemporary social life, highlighting the applied implications of this line of research. Topics covered include the role of insecurity in social communication, social judgments, decision making, group identification, morality, interpersonal behaviour, relationships, attitudes and many applied aspects of social life and politics where understanding the psychology of insecurity is of critical importance. This accessible and engaging book will be of interest to students, researchers and practitioners as a textbook or reference book in behavioral and social science fields, as well as to a broad spectrum of intelligent lay audience seeking to understand one of the most intriguing issues that shapes human social life.

A Friendly Introduction to Abstract Algebra offers a new approach to laying a foundation for abstract mathematics. Prior experience with proofs is not assumed, and the book takes time to build proof-writing skills in ways that will serve students through a lifetime of learning and creating mathematics. The author's pedagogical philosophy is that when students abstract from a wide range of examples, they are better equipped to conjecture, formalize, and prove new ideas in abstract algebra. Thus, students thoroughly explore all concepts through illuminating examples before formal definitions are introduced. The instruction in proof writing is similarly grounded in student exploration and experience. Throughout the book, the author carefully explains where the ideas in a given proof come from, along with hints and tips on how students can derive those proofs on their own. Readers of this text are not just consumers of mathematical knowledge. Rather, they are learning mathematics by creating mathematics. The author's gentle, helpful writing voice makes this text a particularly appealing choice for instructors and students alike. The book's website has companion materials that support the active-learning approaches in the book, including in-class modules designed to facilitate student exploration.

Ah Love Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme."

The problems of constructing covering codes and of estimating their parameters are the main concern of this book. It provides a unified account of the most recent theory of covering codes and shows how a number of mathematical and engineering issues are related to covering problems.

Scientists involved in discrete mathematics, combinatorics, computer science, information theory, geometry, algebra or number theory will find the book of particular significance. It is designed both as an introductory textbook for the beginner and as a reference book for the expert mathematician and engineer.

A number of unsolved problems suitable for research projects are also discussed.

This volume contains the proceedings of the Arizona Winter School 2016, which was held from March 12-16, 2016, at The University of Arizona, Tucson, AZ. In the last decade or so, analytic methods have had great success in answering questions in arithmetic geometry and number theory. The School provided a unique opportunity to introduce graduate students to analytic methods in arithmetic geometry. The book contains four articles. Alina C. Cojocaru's article introduces sieving techniques to study the group structure of points of the reduction of an elliptic curve modulo a rational prime via its division fields. Harald A. Helfgott's article provides an introduction to the study of growth in groups of Lie type, with $\mathrm{SL}_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. The article by Etienne Fouvry, Emmanuel Kowalski, Philippe Michel, and Will Sawin describes how a systematic use of the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz and Laumon help make progress on various classical questions from analytic number theory. The last article, by Andrew V. Sutherland, introduces Sato-Tate groups and explores their relationship with Galois representations, motivic $L$-functions, and Mumford-Tate groups.

Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas.

This series presents some tools of applied mathematics in the areas of proba bility theory, operator calculus, representation theory, and special functions used currently, and we expect more and more in the future, for solving problems in math ematics, physics, and, now, computer science. Much of the material is scattered throughout available literature, however, we have nowhere found in accessible form all of this material collected. The presentation of the material is original with the authors. The presentation of probability theory in connection with group represen tations is new, this appears in Volume I. Then the applications to computer science in Volume II are original as well. The approach found in Volume III, which deals in large part with infinite-dimensional representations of Lie algebras/Lie groups, is new as well, being inspired by the desire to find a recursive method for calcu lating group representations. One idea behind this is the possibility of symbolic computation of the matrix elements. In this volume, Representations and Probability Theory, we present an intro duction to Lie algebras and Lie groups emphasizing the connections with operator calculus, which we interpret through representations, principally, the action of the Lie algebras on spaces of polynomials. The main features are the connection with probability theory via moment systems and the connection with the classical ele mentary distributions via representation theory. The various systems of polynomi als that arise are one of the most interesting aspects of this study."

Within the last decade, semigroup theoretical methods have occurred naturally in many aspects of ring theory, algebraic combinatorics, representation theory and their applications. In particular, motivated by noncommutative geometry and the theory of quantum groups, there is a growing interest in the class of semigroup algebras and their deformations.

This work presents a comprehensive treatment of the main results and methods of the theory of Noetherian semigroup algebras. These general results are then applied and illustrated in the context of important classes of algebras that arise in a variety of areas and have been recently intensively studied. Several concrete constructions are described in full detail, in particular intriguing classes of quadratic algebras and algebras related to group rings of polycyclic-by-finite groups. These give new classes of Noetherian algebras of small Gelfand-Kirillov dimension. The focus is on the interplay between their combinatorics and the algebraic structure. This yields a rich resource of examples that are of interest not only for the noncommutative ring theorists, but also for researchers in semigroup theory and certain aspects of group and group ring theory. Mathematical physicists will find this work of interest owing to the attention given to applications to the Yang-Baxter equation.

This book offers a comprehensive account of not necessarily commutative arithmetical rings, examining structural and homological properties of modules over arithmetical rings and summarising the interplay between arithmetical rings and other rings, whereas modules with extension properties of submodule endomorphisms are also studied in detail. Graduate students and researchers in ring and module theory will find this book particularly valuable.

Over the course of his distinguished career, Nicolai Reshetikhin has made a number of groundbreaking contributions in several fields, including representation theory, integrable systems, and topology. The chapters in this volume - compiled on the occasion of his 60th birthday - are written by distinguished mathematicians and physicists and pay tribute to his many significant and lasting achievements. Covering the latest developments at the interface of noncommutative algebra, differential and algebraic geometry, and perspectives arising from physics, this volume explores topics such as the development of new and powerful knot invariants, new perspectives on enumerative geometry and string theory, and the introduction of cluster algebra and categorification techniques into a broad range of areas. Chapters will also cover novel applications of representation theory to random matrix theory, exactly solvable models in statistical mechanics, and integrable hierarchies. The recent progress in the mathematical and physicals aspects of deformation quantization and tensor categories is also addressed. Representation Theory, Mathematical Physics, and Integrable Systems will be of interest to a wide audience of mathematicians interested in these areas and the connections between them, ranging from graduate students to junior, mid-career, and senior researchers.

The book describes developments on some well-known problems regarding the relationship between orders of finite groups and that of their automorphism groups. It is broadly divided into three parts: the first part offers an exposition of the fundamental exact sequence of Wells that relates automorphisms, derivations and cohomology of groups, along with some interesting applications of the sequence. The second part offers an account of important developments on a conjecture that a finite group has at least a prescribed number of automorphisms if the order of the group is sufficiently large. A non-abelian group of prime-power order is said to have divisibility property if its order divides that of its automorphism group. The final part of the book discusses the literature on divisibility property of groups culminating in the existence of groups without this property. Unifying various ideas developed over the years, this largely self-contained book includes results that are either proved or with complete references provided. It is aimed at researchers working in group theory, in particular, graduate students in algebra.

This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.

* What is the essence of the similarity between linearly independent sets of columns of a matrix and forests in a graph?
* Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph?
* Can we test in polynomial time whether a matrix is totally unimodular?
Matroid theory examines and answers questions like these. Seventy-five years of study of matroids has seen the development of a rich theory with links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical and structural engineering.
This book falls into two parts: the first provides a comprehensive introduction to the basics of matroid theory, while the second treats more advanced topics. The book contains over seven hundred exercises and includes, for the first time in one place, proofs of all of the major theorems in the subject. The last two chapters review current research and list more than eighty unsolved problems along with a description of the progress towards their solutions.
Reviews from previous edition:
"It includes more background, such as finite fields and finite projective and affine geometries, and the level of the exercises is well suited to graduate students. The book is well written and includes a couple of nice touches ... this is a very useful book. I recommend it highly both as an introduction to matroid theory and as a reference work for those already seriously interested in the subject, whether for its own sake or for its applications to other fields." -- AMS Bulletin
"Whoever wants to know what is happening in one of the most exciting chapters of combinatorics has no choice but to buy and peruse Oxley's treatise." -- The Bulletin of Mathematics
"This book is an excellent graduate textbook and reference book on matroid theory. The care that went into the writing of this book is evident by the quality of the exposition." -- Mathematical Reviews

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton's quaternion algebra, Grassmann's outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres. With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today's geometric algebra.

Elementary Methods in Number Theory begins with "a first course in number theory" for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdös-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.