## Recent book reviews

Five recent book reviews

## Invisible in the storm: The role of mathematics in understanding weather

Author(s):
Ian Roulstone & John Norbury
Publisher:
Princeton University Press
Year:
2013
ISBN:
978-0-691-15272-1
Short description:

This book is about the role of mathematics in explaining why it is possible to understand weather and climate, even in the presence of chaos. There are degrees of unpredictability, but there are also many stabilizing mechanisms and, most importantly, there is mathematics to quantify the rules. The authors have done a brilliant work to collect a huge amount of historical information, as well as mathematical information, but keeping always a level in the explanations that makes the text accessible to undergraduate students in the first years, and even to people not so familiar with mathematics. All in all, this is a very interesting and enjoyable reading.

MSC main category:
86 Geophysics
MSC category:
86A10
Review:

This book gives a deep insight of the mathematics involved in the forecast of weather. Being addressed to the general public, with a very leisurely and friendly style, it starts by accounting the history and personalities involved in the developing of the scientific understanding of the weather processes over the Earth.

The author explains the equations that govern the weather, which include seven variables: velocity (of wind), time, pressure, moisture, and density of air. The mathematical bits are separated from the main text and are commented without much detail, not to deter the non-mathematical reader. All names of people involved and the contributions and lives of each of them appear along the way.

The equations are hard to analyze (which is the reason of the difficulty of forecasting), so until the appearance of efficient computers able to do these calculations, it was difficult to do reliable predictions. But the use of mathematics in weather dates back from the beginning of the twentieth century, with the pioneering work of Vilhelm Bjerknes, who analyzed the behaviour of the equations for the vorticity, which involve less number of variables. This together with the improvement of the recollection of empirical atmospheric data at locations, and the improvement of the transmission of data to a central point, allowed to depict the well-known meteorological charts that we are so familiarized to watch on the TV News. These have been used since the beginning of the XX century to predict qualitatively the weather for several days ahead. Nowadays, this is done by the use of computing power, by discretizing the set of differential equations involved and treating them numerically.

Large scale phenomena, like the effect of the rotation of the Earth in the circulation of great masses of air or the formation of cyclones, and more local phenomena, like the movement of clouds or the sea breeze, are treated in the book. The authors also explain the theory of chaos, which says that in non-linear problems, even small inaccuracies in initial data can lead to very large deviations in the evolution of the solution. This is inherent to the analysis of weather, making impossible to get accurate solutions for more than 10 days ahead even with the best of the actual supercomputers.

The second half of the XX century witnessed the raise of pure mathematical methods to analyze the equations of meteorology, together with the simultaneous appearance and use of computers. The study of hydrostatic and geostrophic phenomena gave a way to understand the qualitative behaviour of weather and make sense of the intractability of its non-linearity. On the other direction, there has been a feedback from the studies of meteorology to pure mathematical areas, like the Lorenz attractor appearing in Dynamical Systems.

This book is about the role of mathematics in explaining why it is possible to understand weather and climate, even in the presence of chaos. There are degrees of unpredictability, but there are also many stabilizing mechanisms and, most importantly, there is mathematics to quantify the rules. The authors have done a brilliant work to collect a huge amount of historical information, as well as mathematical information, but keeping always a level in the explanations that makes the text accessible to undergraduate students in the first years, and even to people not so familiar with mathematics. All in all, this is a very interesting and enjoyable reading.

Reviewer:
Vicente Muñoz
Affiliation:
UCM

## Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces

Author(s):
Joram Lindenstrauss, David Preiss, Jaroslav Tiser
Publisher:
Princeton University Press
Year:
2012
ISBN:
9780691153568
Short description:

This book explains the main results known about differentiability of Lipschitz functions $f:X\to Y$, where $X,Y$ are Banach spaces, as well as it presents, with full proofs, new, important results of the authors which were only announced in previous publications.

MSC main category:
46 Functional analysis
MSC category:
46G05
Other MSC categories:
46B20, 49J50
Reviewer:
Daniel Azagra
Affiliation:

## Shaping Space. Exploring Polyhedra in Nature, Art, and the Geometrical Imagination

Author(s):
Marjorie Senechal (ed.)
Publisher:
Springer Verlag
Year:
2013
ISBN:
978-0-387-92713-8 (hbk)
Price (tentative):
39,99 € (net)
Short description:

This is the second edition of the Birkhäuser edition of 1987 that has been given a full makeover. It is a collection of papers by different authors about the definitions and descriptions and how to become familiar with polyhedra by actually building them, about their history, their role in nature and art, but also about the mathematics that are involved.

URL for publisher, author, or book:
www.springer.com/mathematics/geometry/book/978-0-387-92713-8
MSC main category:
51 Geometry
MSC category:
51M20
Other MSC categories:
90C57, 52B20, 68R10, 97K30, 00A99
Review:

As the editor quotes in her introduction "plus que ça change, plus c'est la même chose" since indeed polyhedra are as new as they are old and given the recent evolution in graphs, discrete and computational geometry, combinatorial optimization, computer graphics, a new edition of the previous version (Birkhäuser, 1987) became unavoidable and it resulted in a complete makeover. The format is still the same (the first edition grew out of a 1984 conference), it consists of a collection of essays by different authors about many different aspects related to polyhedra.

The papers are ordered in such a way that they start with elementary, less formal definitions an properties, and suggestions and practical tips about how to actually organize hands-on sessions where children are encouraged to construct the three-dimensional objects. But polyhedra are also followed along their historical and cultural trail from the pyramids in old Egypt and the Platonic solids, till recent developments.

In a second part, appearance and use of polyhedra in art and nature is the the central theme. They lived in the minds of the architects of the pyramids but they also appear in futuristic constructions of modern architecture. Because their graphs have some optimality and stability properties also nature's architect is eager to make use of these structures. Crystals, chemical bindings, cell biology quite often follow the geometrical laws of polyhedral constellations. And of course many artists made 2 of 3-dimensional artwork inspired by these forms.

In part 3, called "polyhedra in geometrical imagination", the contributions become more mathematical. Here we find more general polyhedra, and discussions about molecular stability, dual graphs, Dirichlet tessellations and spider webs, diophantine equations, rigidity, decomposition of solids, etc. The final contribution is a set of 10 geometrical problems that are still (partly) open problems still waiting for a solution.

Although there are 22 papers by many different authors, there is an extensive global index that helps you to find the items you are looking for. The readability of the papers is kept as smooth as possible by collecting notes, remarks and references in a section at the end of the book. Of course the style cannot be uniform since there is a difference between an historical survey, an exposition of how to glue pieces of cardboard together, and a mathematical paper with theorems. However, by the ordering of the papers, the reader grows gradually into the mathematics as he of she is reading on towards the end of the book.

The book is amply illustrated and aiming at a public from 9 till 99. It will be of interest to a very broad public. Form a mathematical side children might be interested in geometrical puzzles and advanced mathematicians may be interested in solving the open problems, and the whole range in between will probably find something interesting of their own taste. But of cause also the non-mathematician will be attracted by these fascinating building blocks in nature, art, science and engineering.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven

## Geometric Realizations of Curvature

Author(s):
Miguel Brozos Vázquez, Peter B Gilkey, Stana Nikcevic
Publisher:
Imperial College Press
Year:
2012
ISBN:
ISBN-13 978-1-84816-741-4, ISBN-10 1-84816-741-5
Short description:

This book is focus on the geometric
realizations of curvature. The authors have organized
some of the results in the literature which fall into
this genre. The findings of numerous investigations
in this field are reviewed and presented
in a clear form, including the latest developments
and proofs.

MSC main category:
53 Differential geometry
MSC category:
53B20
Review:

A central area of study in Differential Geometry
is the examination of the relationship between
purely algebraic properties of the Riemannian
curvature tensor and the underlying geometric
properties of the manifold. The decomposition
of the appropriate space of tensors into irreducible
modules under the action of the appropriate structure
group is crucial. This book is focus on the geometric
realizations of curvature. The authors have organized
some of the results in the literature which fall into
this genre. The findings of numerous investigations
in this field are reviewed and presented
in a clear form, including the latest developments
and proofs.

We recall that, given a family of tensors
$\{T_1,\dots ,T_k\}$ on a vector space $V$,
the structure $\left( V,T_1,\dots ,T_k\right)$ is
said to be \emph{geometrically realizable} if there exist
a manifold $M$, a point $P$ of $M$, and an isomorphism
$\phi \colon V\rightarrow T_PM$ such that
$\phi ^{\ast }L_i(P)=T_i$ where $\{L_1,\dots ,L_k\}$
is a corresponding geometric family of tensor fields on $M$.

The book is organized as follows: In Chapter 1 the authors
introduce some notations and state the main results of the book.
They also discuss the basic curvature decomposition results
leading to various geometric realization results in a number
of geometric contexts. The details and proofs can be found
in the rest of the Chapters. Chapter 2 is devoted to
representation theory and in Chapter 3 some results
from differential geometry are presented. In Chapter 4 and 5
the authors work in the real affine and (para)-complex affine
setting respectively. In Chapter 6 and 7
they perform a similar analysis for real Riemannian geometry
and (para)-complex Riemanian geometry. The results in the
(para)-complex and in the complex settings are presented in
parallel. Finally the authors present a list of the main notational
conventions. Following the list a lengthy bibliography is included.
The book concludes with an index.

Reviewer:
Affiliation:
Departamento de Matemática Aplicada, Escuela Técnica Superior de Arquitectura, UPM, Spain

## What's Happening in the Mathematical Sciences - Volume 8

Author(s):
Dana Mackenzie
Publisher:
American Mathematical Society
Year:
2011
ISBN:
ISBN-10: 0-8218-4999-9, ISBN-13: 978-0-8218-4999-6
Price (tentative):
US\$ 23
Short description:

This is the eighth volume of the series "What's Happening in the Mathematical Sciences". The goal of this book, and of the whole series, is to give account for some recent progress in mathematics. The topics covered in the nine chapters of this book range from the high-dimensional topology to quantum chaos and include applications in computer science, medicine, financial markets, ...

URL for publisher, author, or book:
www.ams.org/bookpages/happening-8
MSC main category:
00 General
MSC category:
00A06
Other MSC categories:
00B15
Review:

This is the eighth volume of the series "What's Happening in the Mathematical Sciences". The series, published by the American Mathematical Society, started in 1993 and its goal is to shed light on some of the outstanding recent progress in both pure and applied mathematics.

The book is divided into nine chapters which present some remarkable mathematical achievements.

The first chapter "Accounting for Taste" describes how Netflix, a movie rental company, offered a million-dollar prize for a computer algorithm to recommend videos to customers. The first year of competition identified matrix factorization as the best single approach. However to factor matrices with unknown elements the winner team had to devise their own strategy combining matrix factorization with regularization and gradient descent. After three years of competition the award was given to the team called BellKor’s Pragmatic Chaos. This is an example of the use of mathematics behind the scenes in everyday life.

The second chapter "A Brave New Symplectic World" is devoted to the conjecture of Weinstein saying that certain kinds of dynamical systems with two degrees of freedom always have periodic solutions. The conjecture was proposed in the late 1970s as a problem in symplectic topology and solved thirty years later by Cliff Taubes. The remarkable thing is that Taube's solution does not stay within the original discipline and borrows some ideas from string theory, developed by physicist Edward Witten.

"Mathematics and the Financial Crisis" described the collapse of the world's financial markets in 2008. The Black-Scholes formula to estimate the value of call options is explained in detail. For some time this formula was almost perfect but a mathematical model is only as good as its assumptions.

"The Ultimate Billiard Shot" deals with the game of outer billiards proposed in 1959 by Bernhard Neumann. The outer billiard table is infinitely large and it has a hole in the center. The question is: Does the table need to be infinitely large? In other words, is there any way a ball that starts near the central region can spiral out to infinity? The answer depends on the shape of the hole. In 2007, Schwartz proved that for certain shapes, an outer billiards shot cannot be contained in any bounded region. The game of outer billiards may seem a bit restricted but is of interest to mathematicians as a toy model of planetary motion.

The fifth chapter, "Simpatient", deals with the controversial recommendation in 2009 by the U.S. Preventive Services Task Force that women aged 40-49 should no longer be advised to have an annual mammogram. A public health panel used six breast cancer model to take this decision. This is an example of the growing acceptance of mathematical models for medical decision-making, at least behind the scenes.

"Instant Randomness" addresses questions of the following type: How long does it take to mix milk in a coffee cup, neutrons in an atomic reactor, atoms in a gas, or electron spins in a magnet? In many systems the onset of randomness is quite sudden. This abrupt mixing behavior is the "cutoff phenomenon", and the time when it occurs is called the mixing time.

Quantum chaos is the topic of the seventh chapter "In Search of Quantum Chaos". In the 1970s and 1980s chaos theory revolutionized the study of classical dynamical systems. In the atomic and subatomic realm chaos seems to be absent. However, there is a gray zone, the semiclassical limit, between he quantum world and the macroscopic world. Mathematicians have recently confirmed the occurrence of quantum chaos in this zone.

Even in the twenty-first century mathematics reveal new phenomena in the ordinary three-dimensional space. This is the topic of the chapter "3-D Surprises". In 2008 and 2009, some new ways to pack tetrahedra extremely densely were discovered. In 2005, two engineers in Hungary discovered a new three-dimensional object similar to a tetrahedron but with curvy sides. It is the first homogeneous, self-righting (and self-wronging!) object.

Last chapter is "As One Heroic Age Ends, a New One Begins". In the 1950s John Milnor constructed 7-dimensional "exotic spheres" which are identical to normal spheres from the viewpoint of continuous topology, but different from the viewpoint of smooth topology. This was the starting point of a new era of high-dimensional topology. But one question, the Kervaire Invariant One problem was open for more than forty years. In 2009 three mathematicians, Mike Hill, Michael Hopkins and Doug Ravenel, answered this question. But this may be just the beginning of what topologists will learn from the new machinery used to solved this problem.

The book is well written and can be of interest to both mathematicians and general public with some background in mathematics. Many pictures and illustrative diagrams are included in the book.

Reviewer:
Antonio Díaz-Cano Ocaña
Affiliation: