Book Description
Beautifully written and elegantly presented, this book is based on 10 lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University. Chung's well-written exposition can be likened to a conversation with a good teacher--one who not only gives you the facts, but tells you what is really going on, why it is worth doing, and how it is related to familiar ideas in other areas. The monograph is accessible to the nonexpert who is interested in reading about this evolving area of mathematics.
Customer Reviews:
Elegant and coherent, but a bit dry and unmotivated.......2006-09-08
This book is elegant and accessible, with a coherent presentation, but is a bit dry and unmotivated. The book would benefit from more applications, which should not be hard to find. I felt like Chapter 8 was the high point of the book, with a discussion of random walks, a matrix-tree theorem and invariant field theory.
The researcher who needs an arsenal of technical results in a clear style will find it here; the student who desires some added perspective may come away somewhat dissatisfied.
Book Description
From reviews of the first edition: "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future." -- Newsletter of the EMS "This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book." -- Mathematica "Each chapter begins with an excellent summary of the content and ends with an exercise section... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists." -- Publicationes Mathematicae Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups. Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond.
Customer Reviews:
Review of Knapp's "Lie groups: beyond an introduction.".......2002-08-13
The short version: this is a superbly written and conceived book; if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen. If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book. It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).
The long version, if you want more convincing or details:
I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.") Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory. Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious. Carefully chosen examples motivate and
clarify the general theory; consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view). But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure. Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding. Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured. This is a fantastic book, and exactly what I was
looking for. Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source.
Customer Reviews:
Very dense but good for the time.......2003-05-14
The author introduces the subject of his book as "the newest approach" to string theory, which he defines in analogy to the point particle theory, as an approach to the calculation of relevant quantities using field theory Lagrangians, instead of "off-shell" S-matrix computations, and which is done in 10 dimensions. The first five chapters of the book is not concerned directly with strings at all, but with the quantization of gauge theories, both pure and with the presence of matter (fermions). The author considers first point particle fields in the light cone gauge. In this gauge the field theory appears nonrelativistic, satisfying a non-relativistic "Schrodinger equation" with an imaginary Hamiltonian. The author then discusses the Yang-Mills theory in the light cone gauge, and derives the free Lagrangian for this theory. This motivates a detailed discussion of the conformal algebra since (nonlinear) representations of the Poincare group model the kinetic term of a free light-cone field theory, and one can obtain these, as the author shows, using the conformal group. He later generalizes to the case where interactions are present, and derives the Feynman rules. The reader can readily see the tension between the demands for covariance and unitarity, that is characteristic of gauge theories. The light-cone gauge is manifestly covariant, but in two less dimensions than the dimension of spacetime the fields are formulated in. This is apparent in the use of the Poincare group ISO(D-1,1), the representations of which are constructed for arbitrary massless and massive theories. The representations are nonlinear in the coordinates, and are constructed from irreducible representations of the SO(D-2) rotation group of the SO(D-1,1) Lorentz group. The conformal group is then SO(D, 2).
In these initial five chapters the reader also gets a detailed overview of the BRST formalism, which is very important in the quantization of gauge theories. This formalism is first introduced in the context of the Hamiltonian formalism, which is manifestly covariant in D - 1 dimensions. This involves as expected a separation of coordinates into space and time with the time components of the gauge fields set to zero. The famous Faddeev-Popov ghosts make their appearance here, since the quantization problem is a problem with constraints. The author gives several reasons for using the BRST formalism, and the reader sees the origin of the Slavnov-Identities, which are generalizations of the amazing Ward identities and are a consequence of the side constraint of unitarity.
The actual consideration of strings first takes place in chapter 6. The large amount of work done by the author in the first five chapters to find a general Poincare- and gauge-invariant action for any collection of fields is finally applied in this chapter and the rest of the book. The idea of viewing strings as 2-dimensional field theories is the main point behind the author's approach. The author quantizes the bosonic string in the light-cone gauge and derives the Poincare algebra, which can be viewed as a specialization of what was done in the first two chapters. This is generalized immediately to the case to the fermionic case by introducing a 2D supersymmetry on the world sheet, in complete analogy with the point particle case in chapter five. In this discussion the reader can see clearly the origin of the requirement that D be equal to 10. A manifestly covariant formalism is then discussed, which is a generalization of the bosonic string and the superparticle of chapter 5. This discussion is interesting in that it shows the origin of the Kac-Moody algebra in the covariant derivatives, and the Virasoro algebra. The BRST formalism is discussed later in the context of the first-quantization of the bosonic string as a constrained problem in the conformal gauge. The Feynman rules for interacting strings are then derived using first the external field formalism, and then using functional integration.
The author gets down to studying string field theory in the context of what was done early in chapter 2 in chapters 10 and 11, namely the light-cone gauge and the BRST formalism, with the goal to include the contributions of the string interactions. As expected, in the free field case the bosonic open strings satisfy a Schroedinger-like equation, and interactions are described by splitting and joining of strings, and as expected from a field-theoretic point of view, the graphs are composed of vertices and propagators. The BRST formalism is done only for the closed string case.The author introduces the reader to how to construct gauge-invariant actions for interacting strings in the last chapter of the book. He is careful to note that a string field theory of interacting strings does not exist, and gives explanations to the difficulties involved in constructing such theories. I have not followed the research on this topic since this book was published, so cannot comment on the present state of atttempts to construct these theories, except for those attempts to give an interpretation of open and closed strings in terms of algebraic topology, C*-algebras, and K-theory. These however do not permit any kind of Feynman rules to be derived. No doubt a perusal of the preprint servers will reveal that this problem has been absorbed in the current emphasis on D-brane and M-theories.
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Methods of Graded Rings (Lecture Notes in Mathematics)
Constantin Nastasescu , and
Freddy van Oystaeyen
Manufacturer: Springer
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ASIN: 3540207465 |
Book Description
The topic of this book, graded algebra, has developed in the past decade to a vast subject with new applications in noncommutative geometry and physics. Classical aspects relating to group actions and gradings have been complemented by new insights stemming from Hopf algebra theory. Old and new methods are presented in full detail and in a self-contained way. Graduate students as well as researchers in algebra, geometry, will find in this book a useful toolbox. Exercises, with hints for solution, provide a direct link to recent research publications. The book is suitable for courses on Master level or textbook for seminars.
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Approximation Theorems in Commutative Algebra: Classical and Categorical Methods (Mathematics and its Applications)
J. Alajbegovic , and
J. Mockor
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover
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ASIN: 0792319486 |
Book Description
Various types of approximation theorems are frequently used in general commutative algebra, and they have been found to be useful tools in valuation theory, the theory of Abelian lattice ordered groups, multiplicative ideal theory, etc. Part 1 of this volume is devoted to the investigation of approximation theorems from a classical point of view. The chapters of this part deal with fields and rings, partly ordered groups, and with multirings and d-groups. Part II investigates approximation theorems from a general, categorical point of view. This part is essentially self-contained and requires only a basic knowledge of category theory and first-order logic. For researchers and graduate students of commutative algebra, category theory, as well as applications of logic.
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Field Theory, Disorder and Simulations
Giorgio Parisi
Manufacturer: World Scientific Pub Co Inc
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ASIN: 9810209649 |
Book Description
This volume is a collection of lectures and selected papers by Giorgio Parisi on the subjects of Field Theory (perturbative expansions, nonperturbative phenomena and phase transitions), Disordered Systems (mainly spin glasses) and Computer Simulations (lattice gauge theories).
The basic problems discussed in the Field Theory section concern the interplay between perturbation theory and nonperturbative phenomena which are present when one deals with infrared or ultraviolet divergences or with nonconvergent perturbative expansions. The section on Disordered Systems contains a complete discussion about the replica method and its probabilistic interpretation, and also includes a short paper on multifractals. In the Simulations section, there is a series of lectures devoted to the study of quantum chromodynamics and a review paper on simulations in complex systems.
The works of Giorgio Parisi have repeatedly displayed a remarkable depth of originality and innovation, and have paved the way for new research in many areas. This personal selection of his lectures and papers, complete with an original introduction by him, undoubtedly serves as a vital reference book for physicists and mathematicians working in these fields.
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Frobenius Splitting Methods in Geometry and Representation Theory (Progress in Mathematics)
Michel Brion , and
Shrawan Kumar
Manufacturer: Birkhäuser Boston
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ASIN: 0817641912 |
Book Description
The theory of Frobenius splittings has made a significant impact in the study of the geometry of flag varieties and representation theory. This work, unique in book literature, systematically develops the theory and covers all its major developments.
Key features:
* Concise, efficient exposition unfolds from basic introductory material on Frobenius splittings—definitions, properties and examples—to cutting edge research
* Studies in detail the geometry of Schubert varieties, their syzygies, equivariant embeddings of reductive groups, Hilbert Schemes, canonical splittings, good filtrations, among other topics
* Applies Frobenius splitting methods to algebraic geometry and various problems in representation theory
* Many examples, exercises, and open problems suggested throughout
* Comprehensive bibliography and index
This book will be an excellent resource for mathematicians and graduate students in algebraic geometry and representation theory of algebraic groups.
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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups:An Algebraic Approach (Mathematics and Its Applications)
L.A. Lambe , and
D.E. Radford
Manufacturer: Springer
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ASIN: 0792347218 |
Book Description
The quantum Yang-Baxter equation is an important equation to solve for applications in physics and topology. This book treats the equation in the context of algebraic systems and as a problem for computer algebra. An up-to-date account of the theoretical foundations of solving the equation is given. The book contains new material which is described in the preface. Audience: The book can be used by graduate students and specialists. Over 200 exercises guide the reader from basic principles to research areas.
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Methods in Ring Theory (Lecture Notes in Pure and Applied Mathematics)
Manufacturer: CRC
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ASIN: 0824701836 |
Book Description
"Furnishes important research papers and results on group algebras and PI-algebras presented recently at the Conference on Methods in Ring Theory held in Levico Terme, Italy-familiarizing researchers with the latest topics, techniques, and methodologies encompassing contemporary algebra."
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Methods in Ring Theory (NATO Science Series C: (closed))
Manufacturer: Springer
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Binding: Hardcover
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ASIN: 9027717435 |
Books:
- Spectral Methods: Fundamentals in Single Domains (Scientific Computation)
- Static Analysis: 13th International Symposium, SAS 2006, Seoul, Korea, August 29-31, 2006, Proceedings (Lecture Notes in Computer Science)
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