Five recent book reviews

Publisher:

Princeton University Press

Year:

2014

ISBN:

978-069-1160-41-2 (pbk)

Price (tentative):

£14.95

Short description:

This is the fourth collection of previously published papers that are brought together in one volume by M. Pitici. The idea is to provide generally accessible papers for the non-mathematician to give insight into the world of professional mathematicians by essays reflecting upon mathematics, its history, and the way it should be taught to future generations.

URL for publisher, author, or book:

press.princeton.edu/titles/10071.html

MSC main category:

00 General

MSC category:

00B15

Review:

This collection, the fourth in a row, was brilliantly selected once more by the editor Mircea Pitici. Mind you, the papers were selected, not published in 2013. The current selection has papers published in the period 2009-2012. As usual, the foreword is provided by a mathematical celebrity. This time it's written by Roger Penrose. Reviews of the two previous collections can be found in this EMS database: see 2011 and 2012.

As in the previous collections we find here a diversity of generally accessible papers that report on issues from mathematics or about mathematics that not only professional mathematicians will (or should be) interested in, also politicians or anybody responsible for science and education are advised to read some of the 20 contributions.

There is an enlightening paper by *Ph.J. Davis* on the drastically changed landscape dominated by computers and multimedia in which mathematics has to operate nowadays. *I. Stewart* in a tribute to A. Turing explains how stripes on animals appear as a consequence of waves, while spots are the result of interfering waves. *T. Tao* illustrates how universal or global laws may result from unpredictable behaviour of a great many individuals. The small world phenomenon discussed by *G. Goth* is a counter-intuitive property of the many individuals populating e.g. social networks. This concept is by now fairly well known by the general public. To find the shortest path visiting several nodes in such a graph is known as a traveling salesman problem, and is hard to solve. It is NP-complete. The problem whether the class of NP problems equals its subclass of easier problems of class P is one of the big open problems in computer science and it shows up in a text by *J. Pavlus*. This is one of the Clay Mathematics Institute millennium prize problems. Solving it would gain you a prize of US$1,000,000.

Large numbers usually entail also randomness and probability and these topics are the subject of several of the papers in this collection. The very basics of randomness are bravely reviewed in a short paper by *C. Seife*. *D. Knuth* illustrates that small random perturbations of harmonics in music are more pleasing. This is a general phenomenon. Perfect symmetry is boring. Slight deviations make symmetry interesting. How come that gamblers become addicted? *S. Johnson* shows that this is because the gambler has the wrong intuitive interpretation of the probabilistic laws like "I've been loosing so long, hence the next time I should win". Notorious errors in the early days of probability theory are placed in an historical context by *P. Gorroochurn*. In financial mathematics *E. Ayache* uses metaphysical arguments to conclude that probability should not be applied to states of the market, but that the market itself should be a category of thought to substitute probability. Probability can analyse the past, but it cannot predict the future.

Geometry is another topic brought to the foreground in several of the texts. *R. Gross* uses the Jerusalem Chord Bridge to discuss Bézier curves, *D. Silver* shows that Dürer's "Painter's Manual" introduces the cuts of a cone. Although Dürer gives a construction method, he mistakingly thought that an ellipse was egg-shaped, i.e. thicker at the bottom than at the top. A somewhat understandable error because the cone is thicker there than at the top of the cut. More of a topological nature are the papers by *K. Delp* about how topological concepts like hyperbolic geometry that were used in design and fashion culture, while *F. and W. Ross* show graphical art that is produced by drawing just one very complicated Jordan curve. Mathematical history is reflected in the paper by *D. Lloyd* about the hoax that was caused by a picture of five neolithic carved stone balls from Scotland that presumably were models for the five Platonic solids. He gives good arguments against this conjecture. More history is represented by *J. Bennett* in his paper about the mathematical (mostly astronomical) instruments that were developed during the 16th to the 18th century.

The paper by *F. Quinn* is about the revolutionary ideas in the period 1890-1930 that have drastically influenced sciences. This revolution caused some bifurcation between core mathematics and the applied vision. The paper bridges the gap between history and education. The latter is another recurrent topic in these collections of "Best Writings on Mathematics". Quinn argues that the cultural attitude towards mathematics in general and to core mathematics in particular that we experience today is largely a matter of neglected public relations. Hence high time to revise our ideas about mathematics education, since that has not fundamentally changed since the Greeks. *A. Sfard*'s contribution is another outspoken plea to drastically change our arguments of why we should teach mathematics and certainly how and what we should teach. Also *E. Maloney and S. Beilock*'s text about `math anxiety', that is shown to exist at a much earlier age than generally accepted is relevant in this context.

Once more this is a marvelous selection of papers about mathematics written by the best. They do not drawn the reader into the mathematical jargon that is only of interest to the mathematical literate. In fact practically no mathematics is needed and formulas are almost completely absent. It is the best possible way of communicating mathematics to the non-mathematician and even the ones suffering from mathematical anxiety will enjoy reading the booklet. Of course this is only a relatively small selection but for the reader longing for more, Pitici gives in his introduction an even longer list of books, papers, websites and blogs that are equally worth reading. Pitici did once more an excellent job, and the result is highly recommended to all with a broad interest in science, history, art, education, philosophy,... which is almost anybody.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Profile Books

Year:

2013

ISBN:

978-1-78125-0990 (hbk)

Price (tentative):

£9.99 (hbk)

Short description:

Not really math but numerology: an encyclopedia of numbers, with entries that give historical facts or that explain the origin of why some numbers play an iconic role in society and culture.

URL for publisher, author, or book:

www.profilebooks.com/isbn/9781781250990/

MSC main category:

00 General

MSC category:

00A99

Review:

Some numbers play a central iconic role in cultures all over the world, like 12 (months, hours), 7 (week days), 10 or 20 (fingers and toes), etc. Other numbers have a more specific cultural or religeous origin, and may differ depending on the community. For example 13 is bad luck number in Western society, while 14 is bad luck in Chinese society.

In this book Rogerson has collected miscellaneous numerological facts in the form of a small encyclopedia. Small in the sense that it is not exhaustive and that it is less than 300 pages, but also the dimensions of book are small (18 x 11.6 x 2.6 cm) like a prayer book or a pocket bible. It grew out of a previous slimmer version, and is currently still growing on the author's website. In fact long lists of names like the 108 names of Krishna, the 50 Argonauts, or the 74 hidden names of Ra etc. should be consulted there. Only the first names on the list are given in the book.

So the collection is a countdown from millions, tens of thousands, ... until two, one, zero. Some numbers have more entries than other, but the general trend is that lower numbers have many more than the hundreds and thousands. Barnaby Rogerson has written several travel guides for Mediteranian countries and he also wrote a biography on the Prophet Muhammad, and other books about the crusades and on Islamic history. With these antecedents it is understandable that most entries have some Islamic, Jewish, or Christian origin. However, there are also many things to learn from entries with an Asian religeous background and of course also from Greek and Norse mythology and much more. You will learn about the story behind all these numbers: why 88 is a lucky number in China, why the USA flag hat 13 stripes, how Dante's Divina Comedia is built on the number 11, the origin of the 4 card suits and the 4 legs of the swastika, and the 3 leaves of the shamrock. It is remarkable that zero, which is a very remarkable number, has only 3 entries. It comes from the Sanskit (Shoonya = void) via the arab (Safira) and the Italian (zefiro) and arrived only late in the Western culture. It is so special among the numbers that it has long been considered as not a number at all.

Of course there is some multiplication and division underlying many of the numbers that organize our lives like 360 which can be divided by 30, 20, 12, 4 which was useful for astronomical computations, and which still survives in the 360 degrees of the circle and the dial of an analog clock. However, it should be clear from the previous description, that there is no math involved here. The mathematical construction of the Fibonacci spiral on the cover is somewhat misleading. No prime numbers, no properties from number theory whatsoever. No stories like Ramanujan's remark about his taxi-cab number 1729. You definitely do find a lot of inspiration for less mathematical number stories to tell.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Boca Raton: CRC Press, Taylor & Francis Group. A Chapman & Hall Book

Year:

2014

ISBN:

978-981-4665-6706-1

Short description:

The first notions of abstract algebra and the basics on factorization and finite group theory are presented in this book in an inquiry based approach. Moreover, some elegant applications on cryptography and NIM games are included. What makes this book different of many others is not the subject matter it covers but how this is done. It does not only present the final elaborate product but it guides the reader to rediscover by herself (himself) how the process flows.

MSC main category:

05 Combinatorics

MSC category:

05C25

Review:

This is an unusual textbook providing an introduction to the basics of abstract algebra: ring theory, group theory and a bit of finite field theory and applications. Of course it gives definitions, statements of theorems and proofs, but what makes it original is that the reader is a very active protagonist of her/his own learning process. Indeed this is a book specially adapted for a self-study of the first steps in algebra. As the authors claim, one does not become a painter after contemplating how a painter works but painting by himself.

The book has been written after a deep reflection about how the learning process works. Remarkably, most objects get into the scene from the particular to the general. For example, the ring of integers Z is not only an example of euclidean domain: the first chapter of the book is devoted to study Z paying special attention to its factorization properties, and so it serves as a model to introduce in subsequent chapters the abstract notions of unique factorization domain (UFD), principal ideal domain and euclidean domain.

The book is organized in units (called investigations). Each unit begins with an illuminating 15 lines presentation explaining the questions the reader should be able to answer after studying it. The units contain very carefully written sections devoted to promote the active participation of the reader: mathematical activities whose difficulty changes in a subtle way, connections between different chapters of the book, and a lot of exercises.

Let us briefly summarize the contents of the book. The first five chapters cover elementary ring theory: irreducibility and factorization in Z and polynomial rings with coefficients in UFD's form the core of this part of the book. Chapter VI deals with elementary group theory. As in the precedent chapters, the main notions are introduced via the most significant and easy examples: Z, Z_n, the groups of n-roots of unity, dihedral and symmetric groups. Cayley's embedding theorem, Abel's theorem on the symplicity of the alternating group, the fundamental theorem of finite abelian groups and Sylow's theorems are the main results in this chapter.

The last chapter is Chapter VII, and it has a rather different nature. As easy applications of the basics on factorization and the theory of finite groups, some fascinating items are introduced: RSA Encryption, Check digits, NIM games and 15 puzzle, and questions on finite fields among others.

The book finishes with two appendices dealing with several preliminary notions and results: mappings and the Well-ordering principle of the natural numbers. Complete solutions to all activities and exercises are available at the authors website.

The book is intended to be studied in a year long course. The authors claim, and they are right, that the students are the main actors in the learning process, and this is why the book is written for them. But it will also be a very useful tool for the teachers of the subject. I am sure that all users of this excellent book will not feel disappointed after its study.

Reviewer:

J.M. Gamboa

Affiliation:

Universidad Complutense de Madrid

Publisher:

Profile Books

Year:

2013

ISBN:

978-1-84668-1998 (pbk)

Price (tentative):

£9.99 (pbk)

Short description:

Ian Stewart, in his well known style, guides us through the history and the developments of the great conjectures of mathematics, the approaches undertaken and the kinds of mathematics that were developed in the attempts (and successes) to solve them. Not only the Goldbach and Riemann conjectures and Fermat's last theorem, but also several others, including the Clay Mathematical Institute millennium prize problems.

URL for publisher, author, or book:

www.profilebooks.com/isbn/9781846683374/

MSC main category:

00 General

MSC category:

00B15

Other MSC categories:

11D99, 11M26, 11P32, 35Q30, 52C17, 58Z05, 68Q15

Review:

The popular science writer Ian Stewart tackles the *Great Problems of Mathematics*, i.e., the conjectures that gained fame because they remained unsolved for many years or that are still unsolved. They are important since they often generate a whole new body of mathematics so that prestigious prizes are rewarded for their solution. Stewart describes the origin, the history, and the development of some of these. It is not easy, but Stewart at least tries, with some success I must say, Stewart largely succeeds in transferring the ideas without the plethora of formulas and difficult mathematical concepts that one would expect. He also illustrates that often a breakthrough is triggered by the development in a seemingly unrelated piece of mathematics.

The first topic is the *Goldbach conjecture* (1742): every integer larger than 2 is the sum of 2 primes. Stewart takes his time to introduce prime numbers, their history and their computation. This pays because several of the subsequent problems are related to prime numbers as well. He explains what has been shown and what is not, how it has been linked to the Riemann conjecture and the open problems left for the present and future generations.

*Squaring the circle* is a problem that dates back to the Greeks and is clearly linked to $\pi$. This is a nice illustration how an obviously geometric problem, is reformulated as an algebraic one, solving polynomial equations, leading to real and later complex analysis (to prove that $\pi$ is irrational and transcendental). Much more recent is the *four colour problem* (1852) Again, the solution is not important for cartography because there are many other reasons to choose a colour for the map, but it started research is networks and graphs, and it has been generalized to colour problems on much more complex topological surfaces. It was finally proved in 1976 by Appel and Haken and it revolutionized the concept of a proof, since it was the first time that a proof relied on the verification by a computer of many cases to which the problem was reduced. Too many to be checked by humans.

Although the original 1611 version of *Kepler's conjecture* appeared in his booklet on 6-pointed snowflakes, it is asking how dense spheres can be packed. Every grocer knows how to mount oranges in a pyramid, but it took 387 years for a proof to be found. Again, a computer was needed to solve the global optimization problem. A formal proof avoiding the computer is still an ongoing project. The *Mordell conjecture* (1922) was proved by Faltings in 1983. It is about Diophantine equations but has a geometric formulation stating that a curve of genus $g>1$ over $\mathbb{Q}$ has only finitely many rational points. Stewart uses this on his path towards *Fermat's last theorem*, since one may start from Pythagoras equation $x^2+y^2=z^2$ with integers to the equation of a circle $(x/z)^2+(y/z)^2=1$ with rational numbers and subsequently to a generalization where the circle is replaced by an elliptic curve.

Although Poincaré got the award of the Swedish King Oscar II in 1887, he did tot really solve the originally posed *three body problem* that was suggested by Mittag-Leffler. Nowadays there are numerical techniques to solve equations with a chaotic solution approximately, rigorous proofs and many questions remain unanswered. The answers are directly related to fundamental questions about the stability of our solar system.

Back to prime number distribution with the *Riemann hypothesis* (1859). Again number theory is lifted to complex analysis in the study of the ζ-function (Stewart needs quite some pages to come to this point). It is explained how this leads to the conjecture that the zeros of the ζ-function are on the critical line $x=1/2$ in the complex plane. This is one of the most famous open problem in mathematics today. It survived Hilbert's 1900 unsolved problems and it is reformulated as one of the Clay millennium problems. Most mathematicians believe it to be true as numerics seem to indicate but a proof is still missing.

The other six millennium problems are the subject of the following 6 chapters, in which the mathematics that Stewart needs become tougher. The *Poincaré conjecture* (1904) was solved by Perelman in 2002, but because it took 8 years for the math community to verify his proof, the introvert and eccentric Perelman, totally disappointed with that situation, has withdrawn from mathematics and refuses all contact with the media. He declined the EMS prize (1996), Fields Medal (2006), and the Clay millennium prize (2010).

The *P/NP problem* is still open and the outcome is uncertain: are hard problems such as the traveling salesman problem solvable with polynomial time algorithms? The answer to this question seems to be NP-hard itself.

Solving the *Navier-Stokes equation* is a problem from applied mathematics. Can one verify that the small changes made by numerical procedures don't miss some turbulent solution because the approximation is not fine enough. In January 2014, Otelbaev claims to have solved this problem. The proof is still under review. The *mass gap hypothesis* relates to quantum field theory of elementary particles. These quantum particles have a nonzero lower bound for their mass even though the waves travel at the speed of light. In relativity theory, the mass would be zero. The *Birch-Swinnerton-Dyer conjecture* is another millennium problem about rational solutions of certain elliptic curve equations. Finally the *Hodge conjecture* connects topology, algebra, geometry and analysis to be able to say something about algebraic cycles on projective algebraic varieties.

Although Stewart tries very hard to introduce the unprepared reader to the problems and the techniques for the latter four problems, the much more advanced mathematical needs make these chapters definitely harder to read than the earlier ones. As a conclusion, he gives his own opinion of what will and what will not be proved in the (near) future. Just in case the reader gave up on the Riemann hypothesis and is looking for inspiration to find another really challenging problem, Stewart provides a list of 12 somewhat less known open questions that are as yet unsolved.

Stewart's entertaining style, his meticulous sketching of the historical context, his sharp analysis of the importance and consequences, his broad insight in the wide spectrum of mathematics and, being a mathematician himself, his understanding of the human behind the mathematician, struggling for solutions and recognition, makes this book a very interesting and highly recommendable read.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Basic Books

Year:

2013

ISBN:

978-0-465-05074-1 (hbk)

Price (tentative):

27,99 USD (hbk)

Short description:

Partly a story of his life, partly an introduction to the essence of his work: the Langlands Program. Frenkel displays his enthusiasm, and love for mathematics and in particular for this "Grand Unifying Theory" of mathematics and quantum physics by taking the reader along on this journey from his first contact with SU(3) to what he now is, a leading mathematician at the forefront of this exciting development in mathematics and quantum physics.

URL for publisher, author, or book:

www.basicbooks.com/full-details?isbn=9780465050741

MSC main category:

81 Quantum theory

MSC category:

81-01

Other MSC categories:

01A60, 11F66, 14D24, 14F05, 17B45, 17B67, 81Q60, 81Rxx

Review:

Ever since he grew up as a boy in Kolomna (Russia), Frenkel was fascinated by elementary particles and quantum physics. It was pointed out to him that to understand these, he should start learning mathematics. So he started reading mathematics in his free time. An obvious choice would be to study at the department of Mechanics and Mathematics (*Mekh-Mat*) of the Moscow State University (MGU). However, back in 1984, his father being Jewish, this was impossible by the ruling anti-Semitism. His second choice was the Institute of Oil and Gas (*Kerosinka*), but he sneaked into the GMU to attend some courses and seminars by Gelfand. On the side he worked on a problem of braid groups proposed by D. Fuchs which resulted in his first paper published in *Funct. Anal. Appl.* at the age of 20. This brought him to study symmetry, (braid) groups and curves over finite fields. Further work brought him straight to the *Langlands Program* that was proposed by *Robert Langland* in 1967 and more formally in 1970. It is based on an earlier idea of *André Weil* who, while imprisoned in 1940 (having a disagreement with the French authorities), wrote a letter to his sister explaining the idea of a mathematical *Rosetta Stone* which would allow to translate results between three seemingly different fields in mathematics into each other: number theory, curves over finite fields, and Riemann surfaces. Exploring this connection has been shown successful by the proof of Fermat's Last Theorem. This connection is the mathematical analog to what the theoretical physicist call the *Grand Unifying Theory* in their study of quantum physics. The mathematical or physical aspects are just two different interpretation of the same theory. So quantum physics is like a fourth column to be added to Weil's *Rosetta Stone*. Frenkel's work with B. Feigin on Kac-Moody algebras came just in time because he got an invitation to spend a semester at Harvard in 1989 at the very time that *perestroika* was emerging. Because of the worsening situation in Russia with an unclear outcome, he decided after his 3 months stay, that it was better not to return home. So he stayed at Harvard where he got his PhD in 1991. Later he became professor of mathematics at UC Berkley. In 2003 he got directly involved in a multi-million DARPA grant to work out more elements of Weil's Rosetta Stone. Since then, his mathematical career is largely devoted to building the bits and pieces of this *Grand Unifying Theory*.

Frenkel makes it crystal clear that he is a passionate lover of mathematics and that his enthusiasm for the *Langlands Program* is immense. This love and passion is what he wants to convey to the reader. The math that most people learn in school is like learning to paint a fence in an art class, while true painting is about creating master pieces like Da Vinci or Picasso did. Mathematics is also a moral duty. Our world is ruled by mathematics that are hidden to most of us. The financial crisis in 2008 was caused by applying mathematics by people that were not controlled in a democratic way because our society does not care about mathematics and most people tend to stay away from it as far as possible. Mathematics should not be restricted to the "initiated few" but it should be shared by everybody. There is nothing more democratic than mathematics. There are no patents for formulas, its a universal language, and a correct formula can only represent truth, the universal truth.

With this conviction, Frenkel wants to transfer not only his love for mathematics but he also wants to show us the beauty of the mathematics that he is devoting most of his life to, and not just the "fence painting" bits. Of course reading this book will not make you a mathematician, but he succeeds by describing his life (at least the part related to his mathematical career) and gradually taking the reader along in his conquest of the mathematics he needed. So he explains symmetries, groups, finite fields, SU(3), manifolds, Galois groups, Lie algebras, sheaves, supersymmetry, strings, branes, etc. All things that are far beyond the low-fi kind of math that one usually finds in popular science books. Of course this is not easy, but I can imagine that his charismatic account will make some readers regret that they are not mathematicians, rather than the usual conviction that mathematics is a natural habitat where only nerds can survive. Many of the more technical details are removed from the main body as (sometimes quite extensive) notes that are collected at the end of the book. For a mathematical reader, they are of course useful, but others may want to skip them and still follow the essence of Frenkel's Conquest of Paradise.

But Frenkel is not only a mathematician. The last chapter of the book is still about mathematics and love, but now revealing the artistic talents of Frenkel. After a visit to Paris, he got the idea to make a film about math. With his neighbour, the author T. Farber, he wrote a screen-play called *The two-body problem* about two men in the South of France, one is a writer, the other a mathematician. They exchange their experiences, their passion for their profession and for women. It was published as a book in 2010. Before starting on the movie project, he wanted to get some cinematographic experience at a smaller scale and decided to produce a short movie. During another visit to France, he joined in with Reine Graves, a young film director. Inspired by a Japanese film of Y. Mishima *Rites of Love and Death* in which a lieutenant commits a ritual suicide together with his wife. Frenkel and Graves imitate the movie more or less. It shows a man (Frenkel) and a women (K.I. May) with in the back a poster with the text istina (Russian for truth). The man tattoos a mathematical formula (the formula of love) on the body of the women. The film is called *Rites of Love and Math*. It was well received, and you will find pictures on the Web of Frenkel teaching in Berkeley, but also where he shows up at the Cannes film festival. In fact by different media, Frenkel tries to transmit the same message: a mathematical formula or mathematics in general can be a thrilling thing of beauty, it can give you goose bumps, one may fall in love with it, it represents the ultimate truth, and it is worth committing your life to. The return you get from it is overwhelming.

One final remark. It is of course a side remark after Frenkel's plea for beauty, but I do not think that the cover design of the book is a success. It shows text in slightly tilted rectangles on a background image that is a detail of Van Gogh's *The Starry Night* painting. The symbolism is obviously well chosen, but it looks terribly chaotic, and I would have preferred a more stylish design representing the mathematical purity and beauty of its contents.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven