Five recent book reviews
The notion of geometric combinatorics is quickly getting a much broader meaning. At present it covers not only a structure of polytopes and simplicial complexes but many further topics and interesting connections to other fields of mathematics. It is worth looking at the contents of this book, which contains written versions of the lecture series presented at a three-week program organised at the IAS/Park City Mathematics Institute in 2004. Counting of lattice points in polyhedra and connections to computational complexity is discussed in lectures by A. Barvinok. Root systems, generalised associahedra and combinatorics of clusters form the topic of the lectures by S. Fomin and N. Reading. Combinatorial problems inspired by topics from differential topology (Morse theory) and differential geometry (the Hopf conjecture) are studied by R. Forman. M. Haiman and A. Woo treat topics around Catalan numbers and Macdonald polynomials (the positivity conjecture). D. N. Kozlov discusses in his lectures chromatic numbers, morphism complexes and Stiefel-Whitney characteristic classes. Lectures by R. MacPherson cover topics such as equivariant homology, intersection homology, moment graphs and linear graphs and their cohomology. R. P. Stanley discusses topics connected with hyperplane arrangements and M. L. Wachs treats poset topology. The book ends with a contribution by G. M. Ziegler on convex polytopes. The book contains an enormous amount of interesting material (including a substantial numbers of exercises).
The aim of this book is to provide a rigorous foundation of the real number system. The first step is a treatment of natural numbers and their properties, which are stated as axioms (only the last chapter outlines the possibility of a construction of natural numbers on a set-theoretical basis). The real numbers are then defined as infinite sequences of decimal digits. At this point, it is possible to introduce the ordering of real numbers and prove the supremum property. The operations of addition and multiplication are first defined for numbers with finite decimal expansion (by shifting the decimal point, the problem is reduced to addition or multiplication of natural numbers); by means of a limit process, they are extended to all real numbers. The book also discusses additional interesting topics, in particular the definition of powers with real exponents, exponential and logarithm functions, Egyptian fractions, computer implementation of arithmetic operations and the uniqueness of real numbers up to isomorphism. The text is elementary and contains numerous remarks on the history of the subject.
The core of the book consists of contributions presented at the Abel bicentennial conference held at the University of Oslo, June 3-8, 2002, commemorating the 200th anniversary of Niels Henrik Abel’s birth. The volume does not contain all the contributions of invited speakers at the conference and not all of the contributors attended the conference. The book contains the opening address of King Harald V and 25 papers devoted to various aspects of Abel’s work. However, the reader can also find here papers treating topics that can be considered as mathematics of the next generation. This includes Manin’s paper on applications of non-commutative geometry in Abelian class field theory for real quadratic fields, Fulton’s paper on quantum cohomology of homogeneous varieties, Kassel’s contribution on Hopf-Galois extensions to non-commutative algebras from the point of view of topology, van den Bergh’s paper on non-commutative crepant resolutions of singularities and Chas and Sullivan’s joint paper on closed string operators in topology leading to Lie bialgebras and higher string algebras. The other contributions describe many aspects of Abel’s work as well as parts of number theory, analysis, algebra and geometry having roots in it. The book contains a huge amount of information of a historical and mathematical nature. It can be recommended not only to those working in fields having roots in one of Abel’s versatile contributions to mathematics, but also to anybody who likes to read how ideas can influence future development.
This book consists of 33 essays trying to show to a nonmathematical community what mathematics and its applications really are, why they are so important and how they influence our day to day life. The essays may be read independently. Thanks to a long experience with mathematics as a researcher and teacher, the author provides many creative discussions and examples, varying from simple to more abstract structures of mathematics. He tries to provide a leitmotif to illustrate the relationship between mathematics and common sense. He writes about more than sixty major topics in mathematics, many of which have significant connections to other branches of knowledge (e.g. cosmology, physics, teaching, logic, philosophy, languages). The reader can find discussions on the nature of logic, numbers, counting and discounting, mathematical thinking, deductions, intuition and creativity, problem solving, conceptions of space, mathematical operations, structures, objects, paradoxes, theorems and proofs, as well as meditations on the influence of the media and wars on the development of mathematics and its position in the society. The author states and answers many interesting questions from many points of view. At the end of each essay the references to material that is both popular and professional are given. The book can be recommended to all who are interested in mathematics and its nature, beauty and role in modern society and science.
This small booklet brings the reader to a strange place called Numberland, where all the (integer) numbers live in a big hotel. An experienced reader soon recognizes that the hotel presented here is nothing other than a version of the famous Hilbert hotel, which was constructed as a tool to illustrate the problems in connection with countability. The small size of the book, the style of writing, a (large) number of illustrations and especially the “fairy-tale-like” language all indicates that the booklet is meant for children, probably around or under ten years of age.
And at this point the reviewer was beginning to get a little unsure as to whether this rather difficult piece of mathematics should be presented and explained to children of that age. Maybe, the children should first become reliably accustomed to the notions of “more than”, “larger”, and even “finitely and infinitely many", and only after that, at a proper age, should they be faced with facts like “there are as many odd integers as there are integers” or “there are as many integer fractions as integers themselves”, which are the main “results” of the book. Also, since the concept of uncountability is not at all addressed (which is correct), the child reader can possibly be driven to the misleading realization that all infinite sets are “equally large”. So there remains a small question if the topic is suitable for children of a “fairy-tale” age. However, if the answer to this question is “yes”, then nothing can stop the reviewer from claiming that the booklet is written in a very nice way, presenting all the ideas clearly (at least for the adult reader) and in a concise yet comprehensive form.