Five recent book reviews

Publisher:

Princeton University Press

Year:

2011

ISBN:

978-0-691-11387-6

Short description:

The book is devoted to present the basic results in Functional Analysis and further advanced topics

related with Harmonic Analysis and Probability theory. It is an excellent book with very clear expositions.

MSC main category:

46 Functional analysis

MSC category:

46-01

Other MSC categories:

46-02 , 46F10

Review:

This book is the fourth volume of a series in Mathematical Analysis by the same authors based on lectures at Princeton

University. The book is written as self-contained and touches also several other branches of mathematical analysis.

As the authors say, besides the basic theorems in Functional Analysis, the main objective is to present the strong link

between Functional analysis and Harmonic analysis. It contains also nice deep introductions to other topics of analysis like

several complex variables, probability theory and Brownian motion. Many classical and modern Function Spaces, very useful in Analysis, are presented in a very natural way with a clear motivation. A special chapter is dedicated to Distributions or generalized functions. Also a long chapter is dedicated to Oscillatory integrals in Fourier Analysis. There is a list of exercises and interesting problems after every chapter.

This is an excellent book with very clear expositions and historic motivations (as the previous three volumes by the same authors). The book is highly recommendable for all those interested in Modern Analysis.

Reviewer:

Francisco L. Hernández

Affiliation:

Universidad Complutense de Madrid

Publisher:

Springer

Year:

2012

ISBN:

978-3-642-28820-3

Price (tentative):

95 euros

Short description:

This volume is dedicated to Stephen Smale on the occasion of his 80th birthday. The volume consists of 18 chapters written by 27 outstanding mathematicians, among which we can find names as S.K. Donalson, M. Freedman or M. Gromov. The editors of the volume have made a great effort to collect articles from such a number of high level mathematicians, as the honored person certainly deserves.

URL for publisher, author, or book:

http://www.springer.com/mathematics/applications/book/978-3-642-28820-3

MSC main category:

00 General

Review:

This volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century. Stephen Smale has done pioneering work in differential topology, global analysis, dynamical systems, nonlinear functional analysis, numerical analysis, theory of computation and machine learning as well as applications in the physical and biological sciences and economics.

The volume consists of 18 chapters written by 27 outstanding mathematicians, among which we can find names as S.K. Donalson, M. Freedman, M. Gromov, Y. Eliashberg, P. Mihailescu, Y. Sinai, J. Mather, A. Iserles, R. Pérez-Marco or Th.M. Rassias, just to name a few. The editors of the volume, Panos M. Pardalos (University of Florida, USA) and Themistocles M. Rassias (National Technical University of Athens, Greece), have made a great effort to collect articles from such a number of high level mathematicians, as the honored person certainly deserves.

The articles contained in the book are research articles of high level in all areas related to the work of Stephen Smale. The list of articles is the following:

- Transitivity and Topological Mixing for C^1 Diffeomorphisms. F. Abdenur, S. Crovisier

- Recent Results on the Size of Critical Sets. D. Andrica, C. Pintea

- The Fox-Li operator as a Test and a Spur for WienerHopf Theory. A. Böttcher, S. Grudsky, A. Iserles

- Kahler Metrics with Cone Singularities along a Divisor. S. Donaldson

- The Space of Framed functions is Contractible. Y. Eliashberg, N. Mishachev

- Quantum Gravity via Manifold Positivity. M. Freedman

- Parabolic Explosions in Families of Complex Polynomials. E. Gavosto, M. Staviska

- Super Stable Kahlerian Horseshoe?. M. Gromov

- A Smooth Multivariate Interpolation Algorithm. J. Guckenheimer

- Bifurcations of Solutions of the 2-Dimensional Navier-Stokes System. D. Li, Y. Sinai

- Arnold Diffusion by Variational Methods. J. Mather

- Turning Washington's Heuristics in Favor of Vandiver's Conjecture. P. Mihailescu

- Schwartzman Cycles and Ergodic Solenoids. V. Muñoz, R. Pérez-Marco

- An Additive Functional equation in Orthogonality Spaces. C. Park, Th. Rassias

- Exotic Heat PDEs.II. A. Prástaro

- Topology at a Scale in Metric Spaces. N. Smale

- Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals. H. Srivastava

- Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry. A. Ungar

This book is a fantastic recopilation of nicely written papers by outstanding mathematicians, and deserves to be in every mathematical library.

Reviewer:

Vicente Muñoz

Affiliation:

UCM

Publisher:

Springer

Year:

2012

ISBN:

978-1-4471-2157-2

Price (tentative):

50 euros

Short description:

The theme of the present book is to build up an analogy between knot theory and number theory, which is baptised as "arithmetic topology". In this analogy, knots correspond to prime numbers. Paralellisms between properties for knots and properties for number fields are given in the 14 chapters of the book, going from basic facts to very recent and deep results in the field. The book is largely self-contained and requires a background in both knot theory and number theory at the graduate level.

URL for publisher, author, or book:

http://www.springer.com/mathematics/numbers/book/978-1-4471-2157-2

MSC main category:

11 Number theory

MSC category:

11Rxx, 11Sxx, 57Mxx

Review:

The theme of the present book is to build up an analogy between knot theory and number theory, which is baptized "arithmetic topology". The book starts with an informative introduction going back to the origins of both knots and arithmetic as is currently understood. It progresses to give a fairly complete account of the basic aspects of knots in three-manifolds before moving to arithmetic rings, number fields, and Galois groups.

In chapter 3, the book starts explaining the parallelisms between knot theory and number theory. where knots correspond to prime numbers. The bridge is the Galois group (Deck transformations) of the covering spaces on the topological side versus the Galois group of a ring or field extension on the arithmetic side. For instance, in chapter 4 it is shown an analogy between the linking number mod 2 of two knots and the Legendre symbol of two primes. This continues in the remaining 10 chapters, where many analogies between the two worlds are shown. The comparisons are slowly increasing in level of difficulty, getting to advanced and recent research in the last chapters. However, definitions are carefully formulated and proofs are largely self-contained throughout the book. When necessary, background information is provided and examples and illustrations are provided. Moreover, every chapter finishes with a conclusion of the objects which are analogous in both sides.

To read this book, it is convenient to have a good level of knowledge in number theory and in knot theory, at least at the graduate level. The analogy is most often used to go from the geometrical to the number theory side, but at some instances the direction is reversed (like in chapter 12 with the Iwasawa Main Conjecture). Most of the results mentioned in the second half of the book are of a very profound nature, and require some familiarity with both subjects.

What the book does not offer is any hint of the reason for the analogies. These are presented ad hoc. But why are there so many similarities? Should one expect the existence of a mechanism to go from knots to primes? Or in the other direction? Or maybe the parallelism is due to the fact that both theories can be expressed as examples of some meta-theory to be developed? (This is not completely outrageous since on the geometrical side the main object used is the fundamental group of the knot complement, which is an algebraic object.) These questions are not analysed or even mentioned. It seems to this reviewer that they are the main philosophical questions in the background.

Reviewer:

Vicente Muñoz

Affiliation:

Universidad Complutense de Madrid

Publisher:

Springer Verlag

Year:

2012

ISBN:

978-1-4614-2406-2

Price (tentative):

39.95 € (net)

Short description:

The first part of the book sketches the life and work of Henri Poincaré. The second part is more technical and discusses some of the publications of Poincaré, with a definite bias towards differential equations and dynamical systems but it also covers topology briefly and includes Poincaré's address to the Society for Moral Education.

URL for publisher, author, or book:

www.springer.com/mathematics/book/978-1-4614-2406-2

MSC main category:

01 History and biography

MSC category:

01A60

Other MSC categories:

37-03, 30F35, 37N05, 35Qxx

Review:

The first part of this book, describes the life and work of Henri Poincaré (Nancy 1854 - Paris 1912). At the age of five he got diphtheria followed by a paralysis from which he recovered. His teachers recognized his great mathematical talent at young age, which helped him later when Poincaré by some misunderstanding didn't perform well at an exam or when by his impatient mind took nontrivial leaps in his solution method, that were not appreciated by his examiners. He entered the *École Polytechnique*, a military school in Paris at the age of 19 and 2 years later the *École des Mines* to become a mine engineer. He presented a dissertation on partial differential equations under supervision of Darboux, Laguerre and Bonnet. He worked briefly as a mine engineer in Vesoul (France) but was soon appointed lecturer at the University of Caen. There we produced his work on automorphic functions, which Poincaré called Fuchsian, a terminology that resulted in a dispute with Klein. His career took off when he was appointed a professorship at the Sorbonne. His way of lecturing, and his scientific contacts (Darboux, Appell, Mittag-Leffer) are described, his marriage and children, and his involvement in the *Dreyfus Affair*. In 1889 Poincaré was awarded the prize of Oscar II, king of Sweden and Norway. During the editorial process of publishing his paper in the *Acta Mathematica*, some clarifications were needed, which actually contained the elements of chaos theory, which was only recognized as the KAM (Kolmogorov-Arnold-Moser) theorem in 1960. The first part ends with an analysis of Poincaré as a philosopher, (his views on mathematics, physics and science in general) and how he was as a person.

Poincaré's interests were very diverse and he made contributions about topics that became later whole new research fields like automorphic functions, qualitative theory of differential equations, bifurcation theory, asymptotic expansions, dynamical systems, mathematical physics, and topology. It is impossible to cover all of these topics in one monograph, but nevertheless, the author did a great job in covering many aspects related to differential equations and dynamical systems in part 2 of this book, leaving out almost all of Poincaré's influence on group theory, algebra, and probability. The chapters in this second part have these titles: Automorphic functions (just an introductory chapter), Differential equations and dynamical systems (this is the most extensive one and discusses Poincaré's 1879 thesis, the chapters of his *Mécanique Céleste*, bifurcation, and the Poincaré-Birkhoff theorem), Analysis Situs (this is a short ne about topology), and Mathematical physics (about several applications of differential equations). In each of these chapters the author summarizes the yeast of Poincaré's texts and gives some background, leaving out most of the unnecessary technicalities, and he adds his own comments and remarks. The remaining two chapters contain Poincaré's address to the *Society for Moral Education* three weeks before his death and short biographical summaries of mathematicians from late 19th - early 20th century.

With this book, Verhulst did a marvelous job in sketching not only the person of Henri Poincaré, but also by restricting to, or rather emphasizing on, the differential equations and dynamical systems, among the diverse subjects that Poincaré worked on, he succeeds very well in communicating the essence of what the theory is about. Admitted, this is not for a mathematical illiterate since there are indeed quite some formulas floating around, (even the first part discusses his contributions although without formulas) but any mathematician or mathematical physicist should be able to understand what is going on, even if this is not his or her core business. Verhulst is clearly very well familiar with the work of Poincaré but he is also well documented about the historical and human facts behind the writings. The many illustrations (mostly portraits of mathematicians) make it so much more a pleasure to assimilate. This is a book obviously interesting for historians of mathematics, but also for any mathematician with some interest in the origin of his or her research field or who want to catch a glimpse of the person and the mind of a genius.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Springer Verlag

Year:

2012

ISBN:

978-3-540-34259-5

Price (tentative):

49.95 € (net)

Short description:

This is the English translation of volume III in the Mathematics and Culture series. The originally Italian version was published in 2003 as proceedings of a conference in 2002 in Venice. It contains texts connecting mathematics and art and culture in general, including architecture, stage plays and movies, music, comic strips, and sculpture. There are also essays about mathematics in connection with Venice (where the conference took place), and a set of papers on mathematics in connection with China (the ICM took place in Beijing in 2002).

URL for publisher, author, or book:

www.springer.com/mathematics/book/978-3-540-34259-5

MSC main category:

00 General

MSC category:

00A60

Other MSC categories:

00B10, 97A40, 01A13

Review:

The Italian versions of this series appeared on a regular basis since 2000 under the title Matematica e Cultura. In fact the project was started after a conference in Venice in 1997. In the English version, 6 volumes are now published. The previous one was Mathematics and Culture VI that was published in 2009. The current volume III was still missing in an English translation. The original Italian book published in 2003 contained the proceedings of the conference, which was held in Venice in 2002, just like the first one in 1997.

As with all the other volumes in the series, this is a collection of papers that highlight the relation between mathematics and culture in the broadest sense. That can go both ways. For example a description of the mathematical tools to digitise music or pictures, or to analyse paintings, or apply group theory to describe mosaic patterns. In these articles the piece of art is created first and mathematics is applied to it. But it can also go the other way: sculptors or painters use mathematical patterns, deliberately or not, to create a piece of art, or an artist can use a computer as a tool and write a program that uses fractal patterns to compose music. The spectrum covered is however much broader than music, sculptures, or paintings. The articles also deal with math education and visualisation, explanation of paradoxes (optical or mental), representation of the 4th dimension, cinema, stage plays, cartoons and comic strips. Such diversity is recurrent in all the volumes of the series. In the present volume, there are two special chapters: one containing several papers related to Venice (where the M&C 2002 conference took place) and one with articles related to Chinese mathematics (since the International Congress of Mathematics in 2002 took place in Beijing). A novelty for this volume III is an audio CD that is included with three short pieces of guitar music to illustrate the paper by Claudio Ambrosini on Escher-like perspectives and music composition.

The conferences on Matematica e Cultura organized on a regular basis and the publication of the book series in English is in my opinion a very good initiative. This is not a collection of deep mathematical results that will advance science immediately, but it is an eye-opener to many who might experience mathematics as an invention to terrorise children at school. Mathematics is everywhere. It is not only in our highly technical, computerised and digitised society, but as this book series illustrates, it is also deeply embedded in the soft sector like arts and culture, where it is least expected, and it is of all ages and of all cultures. It is an excellent tool to raise public awareness of mathematics. It can be easily used by teachers or lecturers as a Trojan horse to conquer the fortress of the less mathematically inclined.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven