Recent book reviews

Five recent book reviews

The Ambient Metric

Author(s): 
Charles Fefferman, C. Robin Graham
Publisher: 
Princeton University Press
Year: 
2011
ISBN: 
978-0-691-15313-1
Price (tentative): 
$55
Short description: 

This monograph is devoted to the study of invariants of conformal classes of (semi-riemannian) metrics in differentiable manifolds. The title, "The Ambient Metric", refers to the object studied in the book. The ambient metric associated to a conformal structure on a manifold was originally introduced by the authors in a short article without complete proofs in 1985.

The book, of technical nature, and carefully written, includes full proofs. It is focused on a very particular problem in geometry. As such, it is addressed to researchers already interested in this area of mathematics.

URL for publisher, author, or book: 
http://press.princeton.edu/catalogs/series/am.html
MSC main category: 
53 Differential geometry
Review: 

This monograph is devoted to the study of invariants of conformal classes of (semi-riemannian) metrics in differentiable manifolds. The title, "The Ambient Metric", refers to the object studied in the book. The ambient metric associated to a conformal structure on a manifold was originally introduced by the authors in a short article without complete proofs in 1985, see [C. Fefferman and C.R. Graham, Conformal invariants, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque, 1985, Numero Hors Serie, 95-116].

The book develops and applies the theory of the ambient metric in conformal geometry. The ambient metric is a (semi-riemannian) metric in $n+2$ dimensions that encodes the information of a conformal class of metrics in $n$ dimensions. The ambient metric has an alternative incarnation as the Poincaré metric, which is a metric in $n+1$ dimensions having the conformal manifold as its conformal infinity.

The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Then Poincaré metrics are introduced and shown to be equivalent to the ambient metric formulation. The special case of self-dual Poincaré metrics in four dimensions is considered, leading to a formal power series proof of LeBrun's collar neighborhood theorem, proved originally using twistor methods, for analytic metrics.

Conformal curvature tensors are introduced by using the ambient metric formulation, and their fundamental properties are established. The book concludes with the construction and characterization of scalar conformal invariants in terms of the curvature of the ambient metric.

The monograph is a text of technical nature, very carefully written, which includes full proofs. it is focused on a particular problem in geometry. As such, it is addressed to researchers already interested in this area of mathematics.

Reviewer: 
Vicente Muñoz
Affiliation: 
UCM

A first course in Loop Quantum Gravity

Author(s): 
Rodolfo Gambini, Jorge Pullin
Publisher: 
Oxford University Press
Year: 
2011
ISBN: 
978-0-19-959075-9
Short description: 

Loop quantum gravity has emerged as a possible avenue towards the
quantization of general relativity. This book is an introduction to this
topic, which is at the level of a graduate student in Theoretical Physics.
The book covers: general relativity, hamiltonian mechanics, Yang-Mills
theories, quantum mechanics, quantum field theories and loop quantum
gravity.

MSC main category: 
81 Quantum theory
Review: 

As it is said in the preface of the book, Loop quantum gravity has emerged
as a possible avenue towards the quantization of general relativity. This is
one of the main streams to attack this outstanding problem in mathematical physics,
although one has to recognize that it has had less impact than string and superstring
theories.

Before starting to read the book, I was delighted to have in my hands what
it seemed to be a leisurely introduction to a topic of such novelty, in
which my background as differential geometer and (mathematical) gauge
theorist could be well suited. The first chapters give overall
introductions in differential geometry, general relativity, semi-riemannian
geometry, always keeping an informal style. So informal that at some points
in touches the incorrectness. However, when reading these chapters, some of
the times even skipping parts (so basic for a mathematician), this was
to be forgiven. Around the second third of the book, physical
(and more interesting) ideas enter the discussion (Yang-Mills theories, quantum mechanics,
quantum field theory), and then the book swaps to the usual "physics
jargon", so extraneous to a mathematician. No much
intention not to lose people not already familiar with the theory
is shown in the text, accompanied by a lack of motivation when introducing physical concepts.
Moreover, not even for the experts the material is
going to be of much use, since details are skipped once and again. The third
part of the book (loop general relativity, loop quantum cosmology) is basically
a review of very technical material. This is where the book touches the topic
announced in the title, but by then all arguments follow a physics line of
reasoning: no rigorous proofs, renormalizations, divergences, etc.
Again, I have gone through a text in mathematical
physics on which my expectations have not been fulfilled. Maybe next time.

Reviewer: 
Vicente Muñoz
Affiliation: 
UCM

A Course in Commutative Algebra

Author(s): 
Gregor Kemper
Publisher: 
Springer
Year: 
2011
ISBN: 
978-3-642-03544-9
Short description: 

This book is an introductory text to commutative algebra which is based on several courses given by the author.

MSC main category: 
13 Commutative rings and algebras
MSC category: 
1301
Review: 

This book is an introductory text to commutative algebra with the idea also of being a guide to the algorithmic branch of the subject. In particular some part of the text is devoted to computational methods. The book is divided in four parts. The first part is devoted to a geometric interpretation of some basic concepts like Hilbert's Nullstellensatz, Notherian and Artinian rings and modules and the Zariski topology. The notion of Krull dimension is discussed in part II. In particular heights of an ideal, localization, integral ring extensions and normal rings, Noether nornalization theorem and Krull's principal ideal theorem are the main topics in this part . The third part is devoted to computational methods and Gröbner bases, and the Buchberger algorithm for computing Gröbner bases is presented in detail with applications to compute elimination ideals, an algorithm to compute the image of a morphism of affine varieties. The notion of Hilbert function and Hilbert series and their relation with dimension Is also algorithmically presented. The last part of the text is devoted to (regular) local rings and local properties of varieties being the Dedekind domains case studied in detailed. As a result of various courses teaching experience, this is a valuable and readable textbook on modern commutative algebra. It contains a huge number of exercises and it appeals to geometric intuition whenever possible. It can be highly recommended for independent reading or as material for preparation of courses.

Reviewer: 
Alejandro Melle Hernandez

Magical Mathematics

Author(s): 
Persi Diaconis and Ron Graham
Publisher: 
Princeton University Press
Year: 
2012
ISBN: 
978-0-691-15164-9
Short description: 

The book under review gives the mathematical foundations of a complete set of magic tricks and juggling. Additonally, a mathematical study of the ancient Chinese book "I Chin" is developed. Finally, a series of interesting studies on historical old books and biographies of celebrated American magicians is given, including Martin Gardner, the author of the foreword.

MSC main category: 
00 General
MSC category: 
00A08
Other MSC categories: 
00A99 , 99A20
Review: 

Faced with the beauty and power of a mathematical tool or result, one is sometimes impelled to say that there is Magic behind Mathematics. This poetic feeling becomes a solid statement when it is read conversely: Magic entertainments are crammed with Mathematics. Under this perspective, the literature of the Mathematics of Magic is steadily growing. In particular, the book under review is an excellent work written by two math professors (P. Diaconis and R. Graham) with a long experience in magical activities based on Mathematics. The reader will find full descriptions of many tricks (with which one can play and amaze friends) as well as, and this makes the thing more exciting, the mathematical principles that support them and their relationship with other situations (from DNA decoding to the Mandelbrot set, to give two instances) where the same principles are found. The final result is an exciting work where anyone with some undergraduate math knowledge will definitely enjoy the effects, secrets and explanations of a selected choice of magic tricks.
The first half of the book is devoted to card tricks, surely the main field where Mathematics have proof their full presence among magicians. In particular, the book analyzes the application of Bruijn sequences, the Gilbreath principle are the properties of shuffles. After this extensive part, the reader finds interesting sections about the Mathematics of the I Ching (the ancient Chinese fortune-telling volume) and the theorems behind good juggling. A third part covers a historic visit of the first contributions on mathematical magic and some short biographies of contemporary celebrated American math-magicians, with the exposition of some of their tricks. Among them, the authors selected Martin Gardner who, in addition, wrote the foreword of the book in April 2010, one month before his passing away.

Reviewer: 
Marco CASTRILLON LOPEZ
Affiliation: 
Universidad Complutense de Madrid, Spain

Fascinating Mathematical People. Interviews and Memoirs.

Author(s): 
Donald J. Albers and Gerald L. Alexanderson, Editors
Publisher: 
Princeton University Press
Year: 
2011
ISBN: 
978-0-691-14829-8
Short description: 

This is the third book in a series. It consists of sixteen biographical sketches of relevant contemporary mathematicians. The sketches are mainly developed through interviews, and illustrated with interesting, often surprising, pictures.

MSC main category: 
01 History and biography
MSC category: 
01A70
Review: 

The editors present here a new collection of profiles of important mathematicians, including two Fields Medal winners. They are five women and eleven men, whose lives and achievements of various and quite different styles make them truly appealing and worth to know. They happen to be mathematicians, which of course is in some cases the main point, but not necessarily. Anyway, their distinct backgrounds, motivations to do Mathematics (sometimes not the first option), their private interests and ways to reach their goals, their views of places and colleagues, mostly explained by themselves with passion, is a reading to recommend. As a hint of the surprises hidden every now and then in the pages of this book, let me mention that among the protagonists there is a practicing dentist at Beverly Hills, a distinguished magician performer, a serious top specialist on The Beatles… Well understood, this is a book to discontinuous reading: one picks it at leisure, takes a look at the contents and chooses what to read. No order is required, nor any systematic dedication, but in the end one sure will read it all..., contrarywise, how to be sure not to miss the interesting parts?

Finally, I quote two excerpts (to be found somewhere in the book):

So much for too much rigor too early? Yes. Take Euler. I think he thought he knew perfectly well what a real number was. He could calculate with them; he could do anything. I don’t think he was in any danger of making mathematical errors because he didn’t understand properly what a real number was.

• For eight centuries, universities have been about teaching. You have to teach your research. Teaching of your research is the alpha and omega of academic life. It’s been true since the first university was established, even if it has sometimes been forgotten.

Reviewer: 
Jesús M. Ruiz
Affiliation: 
UCM