1 comments about Nonlinear Waves and Solitons (Japanese Series).

1. This book was written by one of the founders of the field of integrability. It is very well written.Lots of examples, historical references, and applications. Sections on KdV and Burgers equation read like a novel. Very good discussion of inverse scattering. And there is, of cause, a very good discussion of Toda lattice. Enjoy!

No comments about Analysis of Dirac Systems and Computational Algebra (Progress in Mathematical Physics).

No comments about Handbook of Numerical Heat Transfer.

No comments about Integrable Systems and Algebraic Geometry: Proceedings of the Taniguchi Symposium, 1997 : Rokko Oriental Hotel, Kobe, June 29-July 4, 199Y, Rims, Kyoto University, July 7-11, 1997.

1 comments about A Guide to Monte Carlo Simulations in Statistical Physics, Second Edition.

1. I agree that it covers a lot of topics, many of them are important. They actually include much more topics in the second edition than the first one. However, the authors seldomly discuss one topic more than a page. It's like reading abstracts of papers. So if you already know the stuff, you don't need this book. Just go for some papers (papers are at least up to date). If you don't know anything about Monte Carlo sampling, this book is not going to help you too much. So don't waste your money on this book. Newman's book or Frenkel's book is much better.

No comments about Multidimensional Diffusion Processes (Classics in Mathematics).

No comments about A Mathematical Introduction to Conformal Field Theory (Lecture Notes in Physics).

No comments about Quantum Information with Continuous Variables.

1 comments about Monte Carlo Simulation in Statistical Physics.

1. This is a nice little book written by two experts of the field. This edition is only an expanded version of earlier editions (by addition of two new chapters, the core of the book chapter 1 to 3 hasn't change at all). The book covers monte carlo techniques through various well-known examples (Ising model, random walk, percolation, self-avoiding random walk). I enjoyed reading the first 3 chapters of the book. In particular, chapter 3 guides the readers and gives them the chance to practice what they should have learned in previous chapter (through 53 exercises). The following 2 chapters (chapter 4 and 5) are not as nicely written. Moreover, there are some serious shortcoming in the book. (1) All codes are written in Fortran. While everyone who can program can easily understand the codes, Fortran belongs to the past and could have been ok for physics students during late 80's (first edition) but not for those at 2006. (2) The guide (chapter 3) should have been the last chapter and have covered subjects in chapters 4 and 5 (3) As I mentioned before, chapter 4 and 5 are not well-organized. (4) The book in general stresses too much on finite-size effects. However, it is an important subject and it tells us how we can scale our simulation result to more realistic cases. By my judgement, the book gives wrong impression about the degree of its importance.
   
   I recommend graduate students who are serious about learning monte carlo methods to read Newman and Barkema book (Monte Carlo Methods in Statistical Physics) instead since it provides a broader view about the subject. Although I highly recommend those who are interested in the subject to go through chapter 3. It is fun and very instructive.

2 comments about Loop Spaces, Characteristic Classes and Geometric Quantization (Modern Birkhäuser Classics).

1. Quite clear cohomological approach to line bundles and geometric pre-quantization. Well-developed theory of gerbes. Very interesting chapters on Kaehler geometry of the space of knots and line bundles on loop spaces. The book is self-contained; it starts with definitions and basic theory of complexes of sheaves, hypercohomology, and a concise introduction to the Deligne and Cheeger-Simons cohomologies. My personal favorite is chapter 5, most likely the only readable text on gerbes. I would recommend this book as a textbook for relevant grad courses. The book is also accesible for advanced math and physics majors.



2. Characteristic classes, one of the most abstract and most difficult subjects to teach, are treated in this book at a level that is fairly understandable. The author endeavors to explain how characteristic classes "do their jobs" in the areas in which they are employed, and, even though he does not give an understanding of the foundations of the subject, a reading of the book will give one some helpful guidance in the gaining of such an understanding. In the introduction in particular, the author gives an excellent overview of the history of characteristic classes and explains how the arise in different areas of mathematics. The book is written for the mathematician in mind, but readers interested in applying the theory of characteristic classes, such as high energy physicists, could gain a great deal from the reading of this book.

   In chapter 1, the author overviews the language of sheaf theory and how to construct complexes of sheaves. Although the presentation is somewhat abstract, the author does give some examples of the constructions, such as the exponential exact sequence of sheaves. Using an injective resolution of a sheaf, the sheaf cohomology groups are defined and then shown to be independent of the injective resolution. Using the idea of a double complex, spectral sequences are introduced, along with the concept of sheaf hypercohomology. The later is constructed using an injective resolution corresponding to a sheaf complex. Most interestingly, the author shows how the hypercohomology of sheaves is related to the Cech cohomology. The later is more concrete from an applications point of view, and is one that can be more readily understood by physicists, as well as de Rham cohomology that is introduced later, and is shown to be a resolution of the constant sheaf of a smooth manifold. The Cech cohomology groups are shown to be canonically isomorphic to the de Rham cohomology groups.

   A cohomology theory not so familiar to most is the Deligne cohomology, which is also introduced in chapter 1. This is also called Cheeger-Simons cohomology by some, and has applications in conformal field theory. The presentation here is actually quite good, as the author shows how Deligne cohomology is related to ordinary cohomology via a few examples, and how Deligne cohomology can be used to compare Cech cohomology classes with de Rham cohomology classes. The chapter ends with an overview of the famous Leray spectral sequence.

   In chapter 2, the author goes into the classification of line bundles, basically using the Weil-Kostant theory. When the line bundle has a connection, the author shows that the isomorphism classes of line bundles with connections is related to the second Deligne cohomology group. The Kostant central extensions of the group of symplectic diffeomorphims is also considered, and the author shows how this acts on sections of line bundles. In chapter 3, the author considers first the topology on the space of singular knots in a smooth three-dimensional manifold, which is shown to great surprise to be a Kahler manifold. Not only that, the author further shows it to have a symplectic, complex, and a Riemannian structure.

   The discussion gets considerably more interesting in chapter 4, wherein the author discusses how to generalize the classical result that the second integral cohomology group of a manifold is the group of isomorphism classes of line bundles over the manifold. The goal is to characterize the third integral cohomology group, and the author does this by using the theory of C*-algebras. The result of Dixmier-Douady relating the algebra of compact operators on a separable Hilbert space is shown to give the geometric description of the third integral cohomology group. The section on connections and curvature in this chapter is especially well written because the author explains and motivates well the eventual identification of the Hilbert space as the space of infinitely differentiable functions on the unit circle.

   In chapter 5, things get more complicated, where the Dixmier-Douady theory of sheaves of groupoids is related to the third integral cohomology group. Torsors are introduced as a generalization of principal bundles. Algebraic geometers frequently refer to sheaves of groupoids as "stacks" and the author discusses these and the idea of a gerbe from the standpoint of category theory. The sheaf of groupoids is shown to represent the third integral cohomology group and the author constructs a cohomology class of the sheaf of groupoids using differential-geometric constructions.

   Chapter 6 considers line bundles over loop spaces, with the holonomy of line bundles initiating the discussion. Interestingly, the author shows that Deligne cohomology again plays a role here, in that the holonomy of a line bundle with a connection can be expressed in terms of a transgression map in Deligne cohomology. The line bundle over the loop space of a smooth manifold is constructed using the sheaf of groupoids over the manifold, and is called the anomaly line bundle associated to the sheaf of groupoids. When the (free) loop space is generalized to the space of oriented singular knots, and the manifold is 3-dimensional, the author shows how to obtain a bundle over this space, and shows the relation to geometric quantization. The central extension of the loop group is considered, interestingly, in terms of action functionals, an approach which has its origins in physics.

   The author ends the book with a discussion of the Dirac monopole. This object has been studied in great detail in the literature, but here the author gives an interesting twist wherein he relates the monopole to the Dixmier-Douady sheaf of groupoids over the three-dimensional sphere, and gives an explicit generator of the third integral cohomology group of this sphere. The classical quantization condition then follows naturally.