Five recent book reviews

Publisher:

European Mathematical Society

Year:

2007

ISBN:

978-3-03719-029-6

Price (tentative):

EUR 98

MSC main category:

30 Functions of a complex variable

Review:

This book is the first volume of a comprehensive treatment of Teichmüller theory and its many different facets. There are 14 papers by different authors, divided into four parts. The first part is devoted to the metric and analytic theory (the Weil-Petersson metric, a harmonic map interpretation of a compactification of the Teichmüller space, the Teichmüller metric, Thurston’s asymmetric metric, decorated hyperbolic structure, Hölder distributions, the Grothendieck dessins and Teichmüller disks). Group theory aspects are studied in the second part (mapping class groups and their subgroups, deformations of Kleinian groups and a geometry of the complex of curves). Surfaces with singularities and discrete Riemann surfaces form the topic of the third part. Beautiful relations to quantum physics are discussed in the last part of the book (including quantization theories of the Teichmüller space, lamination spaces, a modular functor from quantized Teichmüller theory and a quantization of the moduli space of irreducible flat PSL(2,R) connections on a punctured surface). The first volume of the handbook already shows an extraordinarily broad spectrum of important and interesting topics related to Teichmüller theory.

Reviewer:

vs

Publisher:

Cambridge University Press

Year:

2007

ISBN:

928-0-521-70876-0

Price (tentative):

GBP 29.99

MSC main category:

16 Associative rings and algebras

Review:

These are the second and third volumes of a long awaited modern treatment of representation theory of finite dimensional algebras, written by some of the leading experts in the area. The first volume dealt with fundamentals of the theory, introducing Auslander--Reiten quivers, tilting theory and classification of representations of finite algebras. The main goal of the second and the third volumes is to study representations of infinite tilted algebras B = End TKQ for a Euclidean diagram Q and an algebraically closed field K and give a complete description of their finite dimensional indecomposable modules, their modules categories mod B and the Auslander-Reiten quivers Γ (mod B).

Volume 2 starts with a chapter on tubes and then develops in detail the structure theory for regular components of concealed algebras of Euclidean type. This is then applied to a complete classification of all indecomposable modules over tame hereditary algebras. In the final chapter, a criterion for infinite representation type is proved and then applied to the Bongartz-Happel-Vossieck classification of all concealed algebras of Euclidean type in terms of quivers and relations.

The first part of Volume 3 culminates in the classification of all tilted algebras of Euclidean type due to Ringel. The next two chapters are dedicated to wild hereditary algebras and to a proof of the Drozd tame-wild dichotomy. In the final chapter, a number of recent results pertaining to the topic are listed without proof; as the authors point out, the extent of the volumes did not allow for presentation of all the contemporary tools (in particular covering techniques and derived categories). Each chapter of both volumes ends with a number of exercises; moreover, there are many examples worked out in detail throughout the text. The volumes are indispensable both for researchers and for graduate students interested in modern representation theory.

Reviewer:

jtrl

Publisher:

European Mathematical Society

Year:

2008

ISBN:

978-3-03719-019-7

Price (tentative):

EUR 58

MSC main category:

46 Functional analysis

Review:

This book develops a theory of wavelet bases and frames for function spaces on various types of domains (such as Euclidean n-spaces) and on related n-manifolds. Basic notation and classical results are repeated in order to make the text self-contained. The book is organised as follows. Chapter 1 deals with the usual spaces on Rn, periodic spaces on Rn and on the n-torus Tn, and their wavelet expansions under natural restrictions for the parameters involved. Spaces on arbitrary domains are discussed in chapter 2. The heart of the exposition is found in chapters 3 and 4, where the author develops the theory of function spaces on so-called thick domains (including wavelet expansions and extensions to corresponding spaces on Rn). In chapter 5 this is completed with spaces on smooth manifolds and smooth domains. In the final chapter, the author discusses desirable properties of wavelet expansions in function spaces (introducing the notation of Riesz wavelet bases and frames). This chapter also deals with some related topics, in particular with spaces on cellular domains. The book is addressed to two types of reader: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions, and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of reader, the preface contains a guide to where one will find basic definitions and key assertions.

Reviewer:

knaj

Publisher:

Imperial College Press

Year:

2008

ISBN:

978-1-84816-031-6

Price (tentative):

GBP 35

MSC main category:

62 Statistics

MSC category:

62-01

Other MSC categories:

60-01

Review:

Information disseminated by newspapers, radio, television and so on brings a lot of data that we can use as support for our decisions. Since the decisions are made in an environment where uncertainty plays an important role, a scientific approach is also based on probability theory. It is easy to misinterpret given data, which leads to the well-known phrase “lies, damned lies and statistics” (L. H. Courtney). As for me, I prefer another phrase: “It is easy to lie with statistics. It is hard to tell truth without statistics” (A. Dunkels). And the author in the introduction correctly writes: “In the 21st century a cultured man should understand something about statistics otherwise he will be led by the nose by those who know how to manipulate statistics for their own ends”.

The book is a very elementary introduction to probabilistic and statistical thinking. The basic ideas are demonstrated on simple examples from everyday life, e.g. how to bet on a horse. There are also non-intuitive problems like the birthday problem and the problem about switching. As a more practical topic we find an application of probability theory to medicine. The elements of statistics presented in the book concern calculations of the mean and variance and normal and Poisson distributions. The book also contains parts devoted to predicting voting preferences, a sampling technique, some statistical tests and building probabilistic models. The author does not assume that the reader is familiar with mathematics. Because of that, an explanation on how to handle expressions like (22)3 is quite long and, similarly, a description of the definition of the number e is also very detailed. Some places in the book deserve critical remarks. The expectation of the normal distribution is called the mean and denoted by the same symbol as the arithmetical mean – this is confusing (p. 117). I cannot agree with the formulation “Assuming that the lifetimes have a normal distribution ...” (p. 120). The lifetimes are nonnegative and so they cannot have a normal distribution. It would be better to say that a normal distribution can be a good approximation to the unknown and perhaps very complicated true distribution of the lifetimes. The book can be recommended to students who are not specialists in mathematics.

Reviewer:

ja

Publisher:

American Mathematical Society

Year:

2007

ISBN:

978-0-8218-3965-2

Price (tentative):

USD 89

MSC main category:

46 Functional analysis

Review:

This book deals with three types of operators on Bergman and Hardy spaces on the open unit disk in the complex plane: Töplitz operators, Hankel operators and composition operators. The main emphasis is on the size of these operators, or more precisely whether they are bounded, compact or belong to Schatten classes. The book starts with a presentation of types of operators on Banach and Hilbert spaces that are studied later along with their basic properties. Then the author proves the classical interpolation theorems and Hölder type inequalities on Lp spaces. After introducing Bergman, Bloch and Besov spaces, the author presents results on the Berezin transform. The next sections are devoted to a study of Töplitz and Hankel operators on Bergman spaces. The presentation proceeds with Hankel operators on Hardy and BMO spaces. The last chapter is devoted to composition operators. The second edition is considerably improved and enriched with recent results. Also, several new proofs are included. The book contains exercises of varying degrees of difficulty and an extensive bibliography.

Reviewer:

jsp