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The study of geodesic flows on homogeneous spaces is an area of research that has recently yielded some fascinating developments. This book focuses on many of these, with one of its highlights an elementary and complete proof by Margulis and Dani of Oppenheim's conjecture.Other features are self-contained treatments of an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; and Ledrappier's example of a mixing action which is not a mixing of all orders. ... Read more

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Includes a rich variety of exercises to accompany the exposition of Coxeter groups

Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

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This book contains papers presented by speakers at the AMS-IMS-SIAM Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, held at Mount Holyoke College in June 1992. One group of papers deals with one aspect of conformal field theory, namely, vertex operator algebras or superalgebras and their representations. Another group deals with various aspects of quantum groups. Other topics covered include the theory of knots in three-manifolds, symplectic geometry, and tensor products. This book provides an excellent view of some of the latest developments in this growing field of research. ... Read more

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A number of eminent mathematicians were invited to Bielefeld, Germany in 1999 to present lectures at a conference on topological, combinatorial and arithmetic aspects of (infinite) groups. The present volume consists of survey and research articles invited from participants in this conference. The contributions are geared to specialists and aspiring graduate and post-graduate students interested in pursuing additional research. ... Read more

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The articles in this volume are based on the talks given at two special sessions at the AMS Sectional meetings held in 2004. The articles cover various topological and asymptotic aspects of group theory such as hyperbolic and relatively hyperbolic groups, asymptotic cones, Thompson's group, Nielsen fixed point theory, homology, groups acting on trees, groups generated by finite automata, iterated monodromy groups, random walks on finitely generated groups, heat kernels, and currents on free groups. ... Read more

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This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra or differential geometry.

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Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.

Customer Reviews (1)

An outstanding introduction to representation theory
Using an old idea of Hunt (P.A.M.S. 1956) this book reachs a beautiful simplicity in itstreatement of conjugacy of maximal tori in a compact connected Lie group G and in the proof that theexponential map of G is onto. As far as I know, mostprevious texts (including Dieudonné's "éléments d'analyse" vol. 5, ISBN 2040009329) used eitherHunt's idea mixed with Hopf-Rinow theorem Introduction to Compact Lie Groups (Series in Pure Mathematics), The structure of Lie groups (Holden-Day series in mathematics),either A. Weil's proof using the Lefschetz fixed point formula,to achieve the same objectivesLie Groups (Graduate Texts in Mathematics), Harmonic analysis on homogeneous spaces (Pure and applied mathematics, 19). The method used in Representations of Finite and Compact Groups (Graduate Studies in Mathematics ; V. 10) are similar to Sepansky's, but I failed to understand some Simons' arguments and I had to switch into Wallach's. The rest of Sepansky's work is fine, but more standard, treating roots and weights as simultaneous eigenvalues ofrepresentations of the algebra of a maximal torus (instead of thinking of them as simultaneous eigenvalues ofrepresentations of the torus itself, as in Lie Groups: Beyond an Introduction, or Representations of Compact Lie Groups.Of course,the book includes a detailed proof of H.Weyl's character formula and it ends with Borel-Weil construction of all irreducible representations of compact semi-simple Lie groups. I think that the theory of representations of compact semi-simple Lie groups is much simpler and easy to understand than the (equivalent) theory of representations of complex semi-simple Lie algebras. For its study, Sepansky's book is brief,simple and self-contained; therefore I fully recommende it. However, the cited books by Knapp and Bump add a rather extensive content on real Lie algebras and their classification. On the other hand, Fegan's little book is still valuable, but it should need a second edition to fill many details and gaps. Finally I will cite a similar book Lie Groups (Universitext) which contains Borel-Weil construction;it's scopeis not much broader than Sepansky's but it relies heavily ontopological group actions, orbits, etc. I think thatSepansky's book can also be compared with the content of "éléments d'analyse" vol. 5 of Jean Dieudonné,for its combination of originality,transparency and (relative) brevity. ... Read more

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A comprehensive and modern account of the structure and classification of Lie groups and finite-dimensional Lie algebras, by internationally known specialists in the field. This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

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This is a well-balanced introduction to topology that stresses geometric aspects.Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group.It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included.New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable. ... Read more

Customer Reviews (3)

Does not deformation retract onto Munkres et al.
This is a wonderfully intellectual, semi-historical approach to classical topology.

Chapter 0 gets some fundamentals out of the way. Chapter 1 is very intriguing and contains lots of ideas. First we are given a taste of the Riemann surfaces of complex analysis. These are complemented by the nonorientable surfaces, and it all leads to the classification of surfaces, which is achieved through the fundamental group and the realisations of surfaces as polygons with identifications, and this in turn leads picturesquely to covering surfaces. These simply and concisely presented ideas provide the seeds for much of the later chapters. The short chapter 2 sets up the two-way connection between topology and combinatorial group theory, which proves fruitful when the fundamental group grows into two chapters of its own (3 and 4). Then follows a sort of supplementary chapter 5 which touches on homology theory (otherwise largely neglected, but with good reason, Stillwell argues) to motivate abelianisation, which is the method we use to formally tell the fundamental groups of all surfaces apart. Chapters 2-5 were a bit slowed down by foundational issues, but now in chapters 6-8 it's all topology all the time. There are nice accounts of the classical theories of curves on surfaces (chapter 6) and knots (chapter 7). In chapter 8 we see how some of our previous ideas carry over to 3-manifolds. But ultimately 3-manifolds are deep water, with the homeomorphism problem being unsolved and all. Neither would it help to move up to 4-manifolds or higher, but at least that's not our fault so to speak because there the homeomorphism problem is in fact unsolvable. The homeomorphism problem and other fundamental problems are essentially algorithmic (i.e., given two spaces, decide whether they are different or the same) so unsolvability (non-existence of algorithms) is indeed a force to be reckoned with, so it is given its own chapter 9, naturally culminating with the unsolvability of the homeomorphism problem.

There are many ways to destroy the soul of topology. Stillwell says in the preface: "In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams." Stillwell protects us from such dangers by his emphasis on low dimensions, his insistence on the fundamental group as the best unifying tool, visualisation and illustrations, and his great respect for primary sources. The latter is reflected in frequent references and in the commented, chronological bibliography, which is very useful.

An introduction well worth reading
Ideas from topology are now manifested everywhere in physics, engineering, computer graphics, and many, many other applications. Consequently, a thorough understanding of topology has become necessary for those who are involved in these applications. This book gives an introduction to "classical" topology that emphasizes the geometric intuition behind the subject, and is thus very suitable for those who need such an understanding. That is not to say that aspiring mathematicians will not gain from the reading of the book. It still maintains a standard of rigor that graduate students in mathematics need to advance in more in-depth courses in topology. The author does not hesitate to use diagrams in the book, which makes it an even better one for those interested in applications. Most of the results he discusses were known in the late 1800's and early 1900's, but they are still important today, especially in physics.

In chapter 0, the author introduces the fundamental concepts in topology with a discussion of the homoemorphism problem. Confining attention to three dimensions and less, the author does mention the impossibility of solving the problem in dimensions greater than or equal to four. He then gives an overview of open and closed sets, continuous functions, identification spaces, and elementary homotopy theory. The building blocks of the main objects he considers in the book, namely simplicial complexes, are discussed in detail. The Haupvermutung is briefly discussed, and a full proof, due to E.E. Moise, of the Jordan curve theorem is given. The proof is the first example of the general approach that the author takes in the book, namely of reducing general topology to combinatorial topology. A brief introduction to algorithms is given, and the author introduces the group theory needed for the rest of the book.

Since he is taking an historical approach in this book, the author begins the study of surfaces with the study of Riemann surfaces. He motivates the ideas of Riemann surfaces, such as branched coverings of the 2-sphere, very nicely, and gives a very understandable theorem for surfaces is proved in detail. In addition, the concept of a universal covering space is described beautifully, and the author shows how to obtain it for orientable surfaces of genus greater than one. The author also gives a brief taste of Fuchsian groups.

Chapter 2 is devoted completely to the group theory of graphs, as a warm up to the study of the fundamental group in the next chapter. The fundamental group is defined to be an equivalence class of maps, and with the exception of the circle, it is calculated using deformation retraction and the Seifert-Van Kampen theorem. The fundamental group of complexes are then calculated in chapter 4, using first a method due to Poincare, and then directly. Some knot theory is introduced, and the Wirtinger presentation of knot groups is discussed.

Homology theory is presented in chapter 5 as an abelianization of the fundamental group, and the abelianization is shown to be independent of the presentation of the fundamental group. The author does not spend much time in homology, arguing (correctly) that in dimensions less than or equal to three the fundemental group contains all the information obtained from homology.

The study of curves on surfaces is the subject of chapter 6, with the contractibility problem studied first using Dehn's algorithm. Some methods for "simplifying" simple curves on closed orientable surfaces by homeomorphisms are also discussed. These techniques could be considered an elementary warm-up to the handle calculus procedures done in higher-dimensional topology. The physicist-string-theorist reader will appreciate the discussion of the mapping class group of the torus.

All of chapter 7 is devoted to knot theory, and this subject, now of enormous importance in physics and computational biology, is treated in great detail here. The braid group is defined, and Artin's solution of the word problem is given.

A very short overview of 3-dimensional manifolds is given in chapter 8, wherein the important concept of a Heegaard splitting is discussed, along with other methods for constructing 3-manifolds. The recognition problem for the 3-sphere, and the famous Poincare conjecture, are mentioned, and the author outlines one method, called shelling a simplicial decomposition, for recoginizing a 3-sphere. He shows however the existence of an unshellable triangulation of the 3-ball (Bing's cube). The author is incorrect though when he states that an algorithm for disproving the conjecture, i.e. an algorithm for enumerating all the 3-manifolds not homeomorphic to the 3-sphere, does not exist. Since the date of publication of this book, such an algorithm has been constructed by Rourke and Sanderson.

The book ends with a discussion of unsolvable problems in combinatorial topology. This is an unusual topic in a book on topology, but given the importance currently of computer algorithms in the growing field of "computational topology" such an inclusion is appropriate and useful. The author discusses Turing machines, Church's thesis, and the Halting problem. The author discusses the unsolvable problems in group theory, and shows how the halting problem can be reduced to the word problem in groups. He leaves as an exercise, using the formalism developed in the chapter on presentations of groups, the problem of showing the homeomorphism problem is unsolvable for closed 4-manifolds.

An accessible introduction to low-dimensional topology.
This book treats (geometric) topology of dimensions two and three together with necessary machinery of group theory. The account is accessible if you have the basic knowledge of general topology and algebra. It well illustrates the inherent difficulties in low-dimensional topology. A mild surprise with this book is that Novikov's proof of the unsolvability of the word problem is fully described. ... Read more

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This is the first book on Abelian Group Theory (or Group Theory) to cover elementary results in Abelian Groups. It contains comprehensive coverage of almost all the topics related to the theory and is designed to be used as a course book for students at both undergraduate and graduate level. The text caters to students of differing capabilities by categorising the exercises in each chapter according to their level of difficulty starting with simple exercises (marked S1, S2 etc), of medium difficulty (M1, M2 etc) and ending with difficult exercises (D1, D2 etc). Solutions for all of the exercises are included. This book should also appeal to experts in the field as an excellent reference to a large number of examples in Group Theory. ... Read more

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The present volume is a new edition of a volume published in 1985, ("Aplicatii ale teoriei grupurilor in mecanica si fízica", Editura Tehnica, Bucharest, Romania). This new edition contains many improvements concerning the presentation, as well as new topics using an enlarged and updated bibliography.
In addition to the large area of domains in physics covered by this volume, we are presenting both discrete and continuous groups, while most of the books about applications of group theory in physics present only one type of groups (i.e., discrete or continuous), and the number of analyzed groups is also relatively small (i.e., point groups of crystallography, or the groups of rotations and translations as examples of continuous groups; some very specialized books study the Lorentz and Poincaré groups of relativity theory). ... Read more

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Generalized Heisenberg groups, or H-type groups, introduced by A. Kaplan, and Damek-Ricci harmonic spaces are particularly nice Lie groups with a vast spectrum of properties and applications. These harmonic spaces are homogeneous Hadamard manifolds containing the H-type groups as horospheres. These notes contain a thorough study of their Riemannian geometry by means of a detailed treatment of their Jacobi vector fields and Jacobi operators. Some problems are included and will hopefully stimulate further research on these spaces. The book is written for students and researchers, assuming only basic knowledge of Riemannian geometry, and it contains a brief survey of the background material needed to follow the entire treatment. ... Read more

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Invited articles by top notch experts

Focus is on topics in representation theory of algebraic groups and quantum groups

Of interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics