Starting from an undergraduate level, this book systematically develops the basics of * Calculus on manifolds, vector bundles, vector fields and differential forms, * Lie groups and Lie group actions, * Linear symplectic algebra and symplectic geometry, * Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.

From the Preface by V. S. VARADARAJAN: "These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them." The papers are arranged chronologically, Volume I collects Harish-Chandra's articles written between 1944 and 1954.

From the Preface by V. S. VARADARAJAN: "These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them." The papers are arranged chronologically, Volume III collects his articles written between 1959 and 1968.

From the Preface by V. S. VARADARAJAN: "These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them." The papers are arranged chronologically, Volume II collects his articles written between 1955 and 1958.

This monograph lays down the foundations of the theory of complex Kleinian groups, a newly born area of mathematics whose origin traces back to the work of Riemann, Poincare, Picard and many others. Kleinian groups are, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can be regarded too as being groups of holomorphic automorphisms of the complex projective line CP1. When going into higher dimensions, there is a dichotomy: Should we look at conformal automorphisms of the n-sphere?, or should we look at holomorphic automorphisms of higher dimensional complex projective spaces? These two theories are different in higher dimensions. In the first case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition. In the second case we are talking about an area of mathematics that still is in its childhood, and this is the focus of study in this monograph. This brings together several important areas of mathematics, as for instance classical Kleinian group actions, complex hyperbolic geometry, chrystallographic groups and the uniformization problem for complex manifolds.

Armand Borel's mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel's work influenced some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan, and Steenrod. In 1992, Borel was awarded the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas." He wrote more than 145 articles before 1982, which were collected in three volumes published in 1983. A fourth volume of subsequent articles was published in 2001. Volume I collects the papers written from 1948 to 1958.

From the Preface by V. S. VARADARAJAN: "These volumes of the Collected Papers of Harish-Chandra are being brought out in response to a widespread feeling in the mathematical community that they would immensely benefit scholars and research workers, especially those in analysis, representation theory, arithmetic, mathematical physics, and other related areas. lt is hoped that in addition to making his contributions more accessible by collecting them in one place, these volumes would help focus renewed attention on his ideas and methods as well as lend additional perspective to them." The papers are arranged chronologically, Volume IV collects Harish-Chandra's articles written between 1970 and 1983.

Armand Borel's mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel's work influenced some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan and Steenrod. In 1992, Borel was awarded the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas." He wrote more than 145 articles before 1982, which were collected in three volumes published in 1983. A fourth volume of subsequent articles was published in 2001. Volume III collects the papers written from 1969 to 1982.

Armand Borel's mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel's work influenced some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan and Steenrod. In 1992, Borel was awarded the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas." He wrote more than 145 articles before 1982, which were collected in three volumes published in 1983. A fourth volume of subsequent articles was published in 2001. Volume II collects the papers written from 1959 to 1968.

Operational Quantum Theory II is a distinguished work on quantum theory at an advanced algebraic level. The classically oriented hierarchy with objects such as particles as the primary focus, and interactions of the objects as the secondary focus is reversed with the operational interactions as basic quantum structures. Quantum theory, specifically relativistic quantum field theory is developed the theory of Lie group and Lie algebra operations acting on both finite and infinite dimensional vector spaces. This book deals with the operational concepts of relativistic space time, the Lorentz and Poincare group operations and their unitary representations, particularly the elementary articles. Also discussed are eigenvalues and invariants for non-compact operations in general as well as the harmonic analysis of noncompact nonabelian Lie groups and their homogeneous spaces. In addition to the operational formulation of the standard model of particle interactions, an attempt is made to understand the particle spectrum with the masses and coupling constants as the invariants and normalizations of a tangent representation structure of a an homogeneous space time model. Operational Quantum Theory II aims to understand more deeply on an operational basis what one is working with in relativistic quantum field theory, but also suggests new solutions to previously unsolved problems.

Profinite groups are of interest to mathematicians working in a variety of areas, including number theory, abstract groups, and analysis. The underlying theory reflects these diverse influences, with methods drawn from both algebra and topology and with fascinating connections to field theory. This is the first book to be dedicated solely to the study of general profinite groups. It provides a thorough introduction to the subject, designed not only to convey the basic facts but also to enable readers to enhance their skills in manipulating profinite groups. The first few chapters lay the foundations and explain the role of profinite groups in number theory. Later chapters explore various aspects of profinite groups in more detail; these contain accessible and lucid accounts of many major theorems. Prerequisites are kept to a minimum with the basic topological theory summarized in an introductory chapter.

This book presents a graduate-level course on modern algebra. It can be used as a teaching book - owing to the copious exercises - and as a source book for those who wish to use the major theorems of algebra. The course begins with the basic combinatorial principles of algebra: posets, chain conditions, Galois connections, and dependence theories. Here, the general Jordan-Holder Theorem becomes a theorem on interval measures of certain lower semilattices. This is followed by basic courses on groups, rings and modules; the arithmetic of integral domains; fields; the categorical point of view; and tensor products. Beginning with introductory concepts and examples, each chapter proceeds gradually towards its more complex theorems. Proofs progress step-by-step from first principles. Many interesting results reside in the exercises, for example, the proof that ideals in a Dedekind domain are generated by at most two elements. The emphasis throughout is on real understanding as opposed to memorizing a catechism and so some chapters offer curiosity-driven appendices for the self-motivated student.

The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.

Perhaps it is not inappropriate for me to begin with the comment that this book has been an interesting challenge to the translator. It is most unusual, in a text of this type, in that the style is racy, with many literary allusions and witticisms: not the easiest to translate, but a source of inspiration to continue through material that could daunt by its combinatorial complexity. Moreover, there have been many changes to the text during the translating period, reflecting the ferment that the subject of the restricted Burnside problem is passing through at present. I concur with Professor Kostrikin's "Note in Proof', where he describes the book as fortunate. I would put it slightly differently: its appearance has surely been partly instrumental in inspiring much endeavour, including such things as the paper of A. I. Adian and A. A. Razborov producing the first published recursive upper bound for the order of the universal finite group B(d,p) of prime exponent (the English version contains a different treatment of this result, due to E. I. Zel'manov); M. R. Vaughan-Lee's new approach to the subject; and finally, the crowning achievement of Zel'manov in establishing RBP for all prime-power exponents, thereby (via the classification theorem for finite simple groups and Hall-Higman) settling it for all exponents. The book is encyclopaedic in its coverage of facts and problems on RBP, and will continue to have an important influence in the area.

Although the study of dynamical systems is mainly concerned with single trans formations and one-parameter flows (i. e. with actions of Z, N, JR, or JR+), er godic theory inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multi-dimensional sym metry groups. However, the wealth of concrete and natural examples, which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. A remarkable exception is provided by a class of geometric actions of (discrete subgroups of) semi-simple Lie groups, which have led to the discovery of one of the most striking new phenomena in multi-dimensional ergodic theory: under suitable circumstances orbit equivalence of such actions implies not only measurable conjugacy, but the conjugating map itself has to be extremely well behaved. Some of these rigidity properties are inherited by certain abelian subgroups of these groups, but the very special nature of the actions involved does not allow any general conjectures about actions of multi-dimensional abelian groups. Beyond commuting group rotations, commuting toral automorphisms and certain other algebraic examples (cf. [39]) it is quite difficult to find non-trivial smooth Zd-actions on finite-dimensional manifolds. In addition to scarcity, these examples give rise to actions with zero entropy, since smooth Zd-actions with positive entropy cannot exist on finite-dimensional, connected manifolds. Cellular automata (i. e.

Providing an accessible approach to a special case of the Rank Theorem, the present text considers the exact finiteness properties of S-arithmetic subgroups of split reductive groups in positive characteristic when S contains only two places. While the proof of the general Rank Theorem uses an involved reduction theory due to Harder, by imposing the restrictions that the group is split and that S has only two places, one can instead make use of the theory of twin buildings.

This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan's famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci's proof of the Poincare-Birkhoff-Witt theorem.

This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo's theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant's structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his "Clifford algebra analogue" of the Hopf-Koszul-Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.

Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.

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This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): "Geometry and Arithmetic of Moduli Spaces of Coverings (2008)" and "Geometry and Arithmetic around Galois Theory (2009)". The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on etale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.

'Et moi, ..., si j'avait Sll comment en revemr, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

This graduate textbook presents the basics of representation theory for finite groups from the point of view of semisimple algebras and modules over them. The presentation interweaves insights from specific examples with development of general and powerful tools based on the notion of semisimplicity. The elegant ideas of commutant duality are introduced, along with an introduction to representations of unitary groups. The text progresses systematically and the presentation is friendly and inviting. Central concepts are revisited and explored from multiple viewpoints. Exercises at the end of the chapter help reinforce the material. Representing Finite Groups: A Semisimple Introduction would serve as a textbook for graduate and some advanced undergraduate courses in mathematics. Prerequisites include acquaintance with elementary group theory and some familiarity with rings and modules. A final chapter presents a self-contained account of notions and results in algebra that are used. Researchers in mathematics and mathematical physics will also find this book useful. A separate solutions manual is available for instructors.

After Pyatetski-Shapiro [PSI] and Satake [Sal] introduced, independent of one another, an early form of the Jacobi Theory in 1969 (while not naming it as such), this theory was given a definite push by the book The Theory of Jacobi Forms by Eichler and Zagier in 1985. Now, there are some overview articles describing the developments in the theory of the Jacobi group and its automor- phic forms, for instance by Skoruppa [Sk2], Berndt [Be5] and Kohnen [Ko]. We refer to these for more historical details and many more names of authors active in this theory, which stretches now from number theory and algebraic geometry to theoretical physics. But let us only briefly indicate several - sometimes very closely related - topics touched by Jacobi theory as we see it: * fields of meromorphic and rational functions on the universal elliptic curve resp. universal abelian variety * structure and projective embeddings of certain algebraic varieties and homogeneous spaces * correspondences between different kinds of modular forms * L-functions associated to different kinds of modular forms and autom- phic representations * induced representations * invariant differential operators * structure of Hecke algebras * determination of generalized Kac-Moody algebras and as a final goal related to the here first mentioned * mixed Shimura varieties and mixed motives.

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.

The Bialowieza workshops on Geometric Methods in Physics are among the most important meetings in the field. Every year some 80 to 100 participants from both mathematics and physics join to discuss new developments and to interchange ideas. This volume contains contributions by selected speakers at the XXX meeting in 2011 as well as additional review articles and shows that the workshop remains at the cutting edge of ongoing research. The 2011 workshop focussed on the works of the late Felix A. Berezin (1931-1980) on the occasion of his 80th anniversary as well as on Bogdan Mielnik and Stanislaw Lech Woronowicz on their 75th and 70th birthday, respectively. The groundbreaking work of Berezin is discussed from today's perspective by presenting an overview of his ideas and their impact on further developments. He was, among other fields, active in representation theory, general concepts of quantization and coherent states, supersymmetry and supermanifolds. Another focus lies on the accomplishments of Bogdan Mielnik and Stanislaw Lech Woronowicz. Mielnik's geometric approach to the description of quantum mixed states, the method of quantum state manipulation and their important implications for quantum computing and quantum entanglement are discussed as well as the intricacies of the quantum time operator. Woronowicz' fruitful notion of a compact quantum group and related topics are also addressed.