NE WS L E T T ER
OF THE EUROPEAN MATHEMATICAL SOCIETY
Feature
History
Interview
ERCOM
Geometric Integration
Mathematics and Physics
Robert Aumann
Max-Planck-Institut Bonn
p. 15
p. 23
p. 29
p. 41
June 2006
Issue 60
ISSN 1027-488X
S E
M
M
E S
European
Mathematical
Society
Recent books
Sums of squares is an important topic in number theory; it represents a melting pot for various methods, extending from elementary ideas to advanced analytic and algebraic techniques.
This book presents many facets of ideas developed in the past
to solve problems connected with the representation of integers as sums of squares. Starting with elementary methods that
have been used since the field was young and ranging to Liouville’s contribution, the book continues through the theory
of modular forms and the analytical methods used to count
the representations. An interesting extension (in comparison
with other books on the subject) is the two chapters devoted to
arithmetic progressions and applications to real life problems.
The first one deals with the theorem of van der Waerden, Roth
and Szemerédi (probably the first book that gives a proof of
this deep result) and the second one covers connections to factorization and applications in the areas of microwave radiation,
diamond cutting and cryptanalysis.
The book is written in a very fresh style and it is essentially
self-contained (e.g. the Riemann-Roch theorem is used but not
proved) and it includes about a hundred interesting exercises of
various levels to test the reader’s understanding of the text. The
book can be recommended to those interested in the development of ideas in the subject and their context. (spor)
S. Oakes, A. Pears, A. Rice: The Book of Presidents 1865–
1965, London Mathematical Society, London, 2005, 157 pp.,
GBP 19, ISBN 0-9502734-1-4
This book concentrates on the history of the London Mathematical Society, which was established in 1865 during the reign
of Queen Victoria and which became one of the most important
scientific societies and well respected in Europe and America.
In the introduction, the authors describe its history using the
lives of its eminent members as a basis. The second part contains
short biographies and scientific evaluations of the presidents of
the London Mathematical Society from 1865 up to 2003 (from
A. De Morgan up to F. C. Kirwan), together with more than
eighty photographs and a list of presidential addresses given at
the conclusion of their respective presidencies. The third part
contains a list of the De Morgan medalists from 1884 (A. Cayley) up to 2004 (R. Penrose) with their photographs and short
biographies. At the end, a list of honorary members from 1867
(M. Chasles) up to 2004 (I. M. Singer) is added. The short glossary explains the history of prizes, sizeships, medals, orders and
“chairs” of the society.
This is an excellent book, which can be recommended to
readers willing to learn about and appreciate the rich history of
the London Mathematical Society and to know which branches
of mathematics were popular in the second part of the 19th
century through to the 20th century. (mbec)
E. M. Ouhabaz: Analysis of Heat Equations on Domains,
London Mathematical Society Monographs, vol. 31, Princeton
University Press, Princeton, 2004, 284 pp., GBP 38,95, ISBN 0691-12016-1
Although the title suggests that heat equations would be the
only topic of the book, the reader soon observes that the book
is devoted to a more general study of the Lp theory of evolution equations associated with non-self-adjoint operators in
divergence form. The author uses the technique of sesquilinear
forms and semigroup theory, which avoids using Sobolev em-
EMS Newsletter June 2006
beddings and therefore does not need smoothness properties
of the boundary. On the other hand, this means that the question of regularity is not addressed by this approach.
One of the aims of the author was to write a self-contained
contribution, which will be useful for a majority of readers. First
we learn some necessary background material from functional
analysis, semigroup theory, sesquilinear forms and the theory
of evolutionary partial differential equations needed to understand the topic. The author proceeds to study semigroups
associated with sesquilinear forms, both for uniformly elliptic
and degenerate elliptic operators. Finally, an even more general
approach is presented. The Lp estimates for the Schrödinger
and wave-type equations are given in the setting of abstract operators on domains of metric spaces. This framework includes
operators on general Riemannian manifolds, sub-Laplacians
on Lie groups or Laplacians on fractals. The book is intended
not only for specialists in partial differential equations but also
for graduate students who want to learn about the sesquilinear
form technique and the semigroup approach to partial differential equations containing second-order elliptic operators in
divergence form. (mrok)
A. Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics 6, European Mathematical Society, Zürich, 2004, 285
pp., EUR 48, ISBN 3-03719-010-8
A lot of standard notions known from Riemannian geometry
have their counterparts for metric spaces. Busemann gave a
definition of a nonpositively curved metric space using convexity properties of the distance function. It is also possible to
study geodesics on a metric space. This book offers a systematic
description of these points. In the first part, the author reviews
basic notions about metric spaces (lengths of paths, length spaces and geodesic spaces, and distances). The second part of the
book studies questions related to convexity in vector spaces.
In the last part of the book, the author discusses Busemann
spaces, locally convex metric spaces, their convexity properties
and some further questions (including properties of convex
functions, isometries, asymptotic rays and visual boundaries).
An important role is played in the book by suitable examples,
in particular by the Teichmüller space. Prerequisites needed to
read the book are modest (basic facts on hyperbolic space and
Teichmüller space). The book can be of interest for mathematicians working in analysis, geometry and topology. (vs)
H.-J. Petsche: Grassmann, Vita Mathematica, vol. 13, Birkhäuser, Basel, 2006, 326 pp., EUR 58, ISBN 3-7643-7257-5
This book (written in German) describes the life and mathematical achievements of Hermann Günther Grassmann (1809–
1877). In the first chapter, the author presents Grassmann’s
life from his childhood up to his death, a life played out on the
background political, social and cultural situation in Germany
during the first half of the 19th century. His family, his studies
at the gymnasium in Stettin and his studies at the University of
Berlin are described in a lot of detail. It may be surprising that
Grassmann had no formal university training in mathematics;
he took courses on theology, classical languages, philosophy
and literature to become a minister in the Lutheran church in
Stettin. After completing his studies in 1830, he became a teacher. After one year of mathematical studies, he took an examina-
61
Recent books
tion to become a teacher of mathematics at gymnasium level.
But his works and his knowledge were not sufficient, so he obtained permission to teach at a lower gymnasium level only.
The author shows Grassmann’s pedagogical activities and
his mathematical production during the two periods (1830–
1840, 1840-1848) during which his most important works were
written. The second chapter describes Grassmann’s mathematical and philosophical academic background. His father was an
excellent professor of mathematics and physics in Stettin, writing several textbooks on mathematics, mineralogy and physics.
His younger brother Robert also became a teacher of mathematics and he collaborated with Hermann on some projects
and publications. The influence that F. D. E. Schleiermacher (a
great philosopher, theologist, politician and teacher) had on
the development of Grassmann‘s philosophical thinking is described here.
The third chapter gives a brief survey of the mathematical
achievements from the 17th century up to the 19th century that
could have influenced Grassmann’s mathematical thinking and
methods. His important publications are very carefully analysed. Grassmann’s contributions to linear algebra, algebraic
forms, the theory of algebras, differential geometry, analysis and
number theory are presented. The fourth chapter explains the
philosophical principles and the background of Grassmann’s
most important and inspired monograph ‘Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik’.
The book contains many interesting pictures, photographs,
reproductions and notes. It will be very helpful for historians
and philosophers of mathematics, for teachers at universities
and secondary schools, and students as well as researchers in
mathematics and history. It can be recommended to people who
are interested in the roots of modern mathematics. (mbec)
M. Rédei, Ed.: John von Neumann: Selected Letters, History
of Mathematics, vol. 27, American Mathematical Society, Providence, 2005, 301 pp., USD 59, ISBN 0-8218-3776-1
This book is a collection of selected letters written by John
von Neumann (1903–1957) to his colleagues, friends, government officials, etc. The book starts with a short description of
von Neumann’s life, career and results in mathematics (logic,
foundations of mathematics, theory of operator algebras and
unbounded operators), physics (quantum mechanics and ergodic theorem), computer science and game theory. His letters
are arranged in alphabetic order of recipient (we find included
here the names G. Birkhoff, P. Dirac, K. Gödel, P. Jordan, I. Kaplansky, E. Schrödinger, E. Segre and N. Wiener). Neumann’s
original text, punctuation and writing style are carefully maintained. In some cases when typographical errors and misspellings would lead to misunderstanding, the reader will find helpful footnotes.
The letters provide readers with a glimpse of Neumann’s
thinking about mathematics, physics, computer science, management, education, politics and war. Some of them contain
technical problems or difficult topics from mathematics and
physics, and they are not easily understandable. Some others
concern daily or general problems and would be understandable to a broad cross-section of the public. At the end of the
book, the reader can find biographical notes on the recipients
of Neumann’s letters (69 scientists, politicians, educational reformers, publishers, etc.), a list of references and a list of Neu-
62
mann‘s publications mentioned in the letters and in the introductory comments. John von Neumann was one of the most
influential mathematicians of the twentieth century. The reader
will find in the book a description of the background and the
development of modern science and many interesting notes on
leading scientists and their works. (mbec)
V. Scheidemann: Introduction to Complex Analysis in Several
Variables, Birkhäuser, Basel, 2005, 171 pp., EUR 25, ISBN 37643-7490-X
The theory of several complex variables is a broad, beautiful
and (nowadays) classical part of mathematics. There are several
excellent books available that describe the theory in full detail.
However, what is often needed for students and researchers
who come from other fields of mathematics is an introduction
to the basic properties of functions of several complex variables
without going into the full and difficult theory. This is what the
reader can find here.
The book starts with the basics (definitions, the Cauchy integral formula, properties of rings of holomorphic functions,
power series expansions and Reinhardt domains). Properties
of the Dolbeault complex are used in the proof of a version of
the Hartogs theorem. A chapter is devoted to the implicit and
inverse function theorems, the Riemann mapping theorem and
to properties of biholomorphic maps. Analytic continuation,
domains of holomorphy, holomorphically convex domains, the
Bochner theorem and the Cartan-Thullen theorems are treated
in two chapters. The book also includes a description of basic
properties of analytic sets and the proof of the Nullstellensatz
for principal ideals. Hence basic features of the theory are introduced and illustrated in a relatively small space (of course,
difficult parts of the theory are not treated here). The book contains a lot of examples and exercises. (vs)
J. Shao: Mathematical Statistics: Exercises and Solutions,
Springer, Berlin, 2005, 359 pp., EUR 39,95, ISBN 0-387-24970-2
This book is a companion to the author’s textbook ‘Mathematical Statistics’ (2nd ed., Springer, 2003), which contains over 900
exercises. This collection consists of 400 exercises and their solutions. Most of the exercises (over 95%) are introduced in the
cited textbook. The reader should have a good knowledge of
advanced calculus, real analysis and measure theory. The book
is divided into seven chapters: 1. Probability theory (measure
and integral, distribution functions, random variables), 2. Fundamentals of statistics (sufficiency, risk functions, admissibility,
consistency, Bayes rule), 3. Unbiased estimation (uniformly
minimum variance unbiased estimators, Fisher information,
U-statistics, linear models), 4. Estimation in parametric models
(conjugate priors, posterior distributions, minimum risk invariant estimators, least squares estimators, maximum likelihood
estimators, asymptotic relative efficiency), 5. Estimation in
nonparametric models (Mallows’ distance, influence function,
L-functionals, Hodges-Lehmann estimator), 6. Hypothesis test
(uniformly most powerful tests, likelihood ratio tests), 7. Confidence sets (Fieller’s confidence sets, pivotal quantity, uniformly
most accurate confidence sets).
The collection is a stand-alone book. It is written very rigorously and solutions are presented in detail. It can be recommended as a source of solved problems for teachers and students of advanced mathematical statistics.
EMS Newsletter June 2006