Five recent book reviews
This book offers an introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory. The subject is first developed following the classical approach analogous to manifold theory. Then the description branches and it includes a useful description of orbifolds by means of groupoids, as well as many examples in the context of algebraic geometry. Classical invariants (e.g. the de Rham cohomology and bundle theory) are developed for orbifolds and a careful study of orbifold morphisms as well as orbifold K-theory and its twistings is provided. The heart of the book is in the penultimate chapter, which contains a detailed description of the Chen-Ruan cohomology, which introduces a new cup product for orbifolds and has had significant impact over the past few years. The final chapter includes explicit computations for certain interesting examples.
This book, the 36th in the Knots and Everything series, ably illustrates the wide range of research involving knots. It provides a rich path through the four ‘countries’ of the knot world: physics, life sciences, numerical simulations and mathematics. There are different countries for various tastes but all are visited with the theory of knots as a companion. It can thus be seen that particle physics returns to knots – as Thomson did by the end of the 19th century with his atomic model – which can be associated with electric flux tubes used for classifying some hadronic states. In plasma physics, it appears that knotted magnetic flux tubes are much more stable than unknotted single loops. Physics is also involved in modelling polymers using knot energies. But it is in life sciences that most of the applications in this book are proposed. DNA knots and protein folds are discussed using various approaches such as Monte-Carlo simulations, microscopy, topological concepts and thermodynamics. One of the most amazing examples is illustrated by the pictures of the uncommon formation of incredibly knotted umbilical cords. Using atomic force microscopy, DNA knots can be observed. It is also known that more complex (i.e. more compact) knots migrate in a gel faster than simple knots, thus explaining why gel electrophoresis can separate knotted DNA according to knot type. On the other hand, knot type can be used to measure the effect of chemical composition on polymer shape. In fact, many applications are devoted to polymers: topological constraints can be related to anomalous osmotic pressure, entanglements to elastic properties, etc. Polymers appears as one of the preferred subjects for stimulating numerical simulations of knot configurations and developing algorithms to investigate properties such as ropelength, i.e. the quotient of the knot length by its thickness.
The book is stimulating because it provides many different types of analysis; analytic computations, numerical simulations and new concepts are discussed with different backgrounds. The diversity of the domains covered occasionally makes the book quite difficult since there are undoubtedly fields that the reader will encounter in which they are not specialized. For instance, there are large gaps between quantum chromodynamics, topological constraints, minimal flat knotted ribbons and gel-electrophoresis. Once this difficulty is left behind (it is always possible to consult the bibliography to obtain more details), the main interest of the book is to provide a wide range of approaches to investigating knots and, consequently, to familiarize the reader with topological invariants, numerical techniques and other statistical techniques according to their own taste.
Since it is devoted to a large number of approaches and applications, this book is recommended to anyone interested in using knots in applied science. Many techniques and concepts are discussed and there is likely to be one for opening a “breach” in the reader’s problem. Among others, there is an interesting open problem concerning the application of knot theory to open strings. Such investigations could have many implications in applied science where closed loops are often hard to identify. The book already provides stimulating approaches to this problem, leading to the concept of knottedness and minimal flat knotted ribbons. It is definitely not an introductory book but provides a nice opportunity to be introduced to advanced research and applications in knot theory. One of its main interests is the intermingling of different research areas such as solid state physics, statistical physics and biophysics. Polymer chains are simply disordered knots that can be viewed on a lattice investigated using statistical properties. Thus thermodynamics and topology can be used together to help understand the complexity of underlying knots and links. What are the possible knots matching a given constraint? Numerical methods can be used to produce various knot types and compute some properties (e.g. ropelength, topological entropic force, average crossing number and probability of knotting). It is interesting to see that crossing numbers can be related to complexity measure and that knots can also inspire statistical approaches. In order to do that, special attention is paid to knots with a large number of crossings. The book offers a nice bridge between various topics too often considered as isolated islands.
At a first glance, this nice, short book is comparable to other brief texts of a similar vein. Its main purpose is to introduce the reader to the basics of algebraic topology and in particular to homology theory and its applications (which is described in depth - about three-quarters of the book is devoted to it) and homotopy theory.
The book begins with homology theory, which is introduced from the geometric approach of simplicial homology. In fact, whenever it is possible within the text, results and proofs are presented through a geometric vision and intuition. Basic and classical properties of simplicial homology together with some applications are then presented. After that, the axiomatic approach to homology theory is described together with a sketch of the proof of uniqueness of homology theory on polyhedra. Then, the author introduces cellular homology and highlights the computational properties of this approach. The chapter also contains the Lefschetz fixed point theorem, a brief introduction to homology with coefficients, and a quick description of some elements of cohomology theory. This introduction to homology theory finishes with a description of Poincaré duality on manifolds.
The basics of homotopy theory are then presented in very brief terms. Starting with the definition of the fundamental group, its computation is described via the Van Kampen theorem applied in different situations. A very short introduction to higher homotopy groups and the action on them by the fundamental group is then followed by an equally extensive presentation of the long exact sequence of a fibre bundle. The chapter is completed with a compendium of the basic properties of covering spaces.
But, as mentioned above, all of this follows from a quick first look at the book under review. After a careful reading, though, this short piece reveals many insights from which different readers may benefit.
The mathematical beginner, perhaps an undergraduate, will find a very intuitive and geometrical, yet formal and rigorous, approach to homology theory. The classical combinatorial difficulties to understanding simplicial homology (a good example being the proof of the simplicial approximation theorem) are overcome with a coherent, geometrical and logical exposition.
The smooth and clear explanation at the beginning of the text of how the abstract concepts of categories and functors are going to be used later on, as well as the geometrical sketch of the proof of the uniqueness of homology theory on polyhedra and the introduction to cellular homology for computational purposes, are good examples of this attempt to bring non-trivial concepts to beginners.
For the graduate student, or the outsider to algebraic topology with some mathematical knowledge and with interests in other branches, this book is also of help. On one hand, some aspects of algebraic topology are presented in a not commonly used approach (for instance, the computation of the fundamental group of the complement of some knots). On the other hand, the use of the basics of simplicial homology theory to attain deep results on more geometrical environments are given in the same geometrical but rigorous language. Here follows an example of this:
After the introduction of the degree of a self map on a manifold, and with the few tools developed at that point, the author readily presents the homotopy classification of immersions of the circle into the plane and the fundamental theorem of algebra, and shows that a vector filed on the 2-sphere has a singular point.
Finally, this text could also be of use for the expert from a teaching point of view. I am sure that any experienced mathematician could find here new ways and tips, at least of an expository nature, of presenting these classical topics in the classroom.
Having said that, there are two more aspects of the text that should be remarked upon, as the author gives them special attention. That is, the exercises and computability.
In fact the whole book is scattered with many exercises of differing scales of difficulty and whose purposes vary. Some of them are provided to encourage and puzzle the independent reader as well as to make them evaluate their knowledge. Others are included to fill gaps left on purpose either to complete proofs of stated theorems or to establish results needed to make the book self-contained. In all cases hints and/or answers to the problems are presented at the end of the text.
On the other hand, concerning computability, by giving explicit and far from complicated algorithms the author points out that homology groups and fundamental groups can be explicitly calculated for many spaces found in nature.
Published in this issue of Philosophical Transactions are the results of a two-day debate held by a group of mathematicians, computer scientists and philosophers, organized in October 2004 by the Royal Society. It contains the talks given at that meeting together with discussions, questions and comments of the participants. Here is a sample of its contents:
• Computing and the culture of proving, by D. MacKenzie
• The challenge of computer mathematics, by H. Barendregt and F. Wiedijk
• What is a proof?, by A. Bundy, M. Jamnik and A. Fugard
• Highly complex proofs and implications of such proofs, by M. Aschbacher
• Skolem and pessimism about proof in mathematics, by P.J. Cohen
• The mathematical significance of proof theory, by A. Macintyre
• The justification of mathematical statements, by P. Swinnerton-Dyer
• Pluralism in mathematics, by E.B. Davis.
Mathematical proofs rank among the highest peaks of human thought but their complexity and variety have increased continuously along the years. Until the middle of the last century it was commonplace to say that a rigorous mathematical proof consisted of a chain of logical steps whose correctness could be checked, in due time, by a person in possession of the appropriate training. Euclid´s Elements contains numerous examples and time, as was correctly stated by Hardy, has not been able to add a single wrinkle to the freshness of their beauty and precision. But the nature of proof has not been static and mathematicians have created new and more powerful strategies, involving new concepts and tools, which provide us with greater freedom and power of reasoning.
Some of these proofs involve long chains of thoughts in an indirect and complicated manner. An example is given by Carleson´s theorem about the almost everywhere convergence of the Fourier series of a square integrable function. Another is the proof obtained by A. Wiles of Fermat´s last theorem. More recently we have the solution of Poincare´s conjecture by G. Perelmann. They are good examples of rigorous proofs that need very many ingenious steps, and which are also deep, elegant and beautiful. But they are so complex that it is doubtful that there exists a single mathematician who would be able to verify the three of them in a reasonable amount of time.
A special treatment deserves the classification of finite simple groups, the proof of which is scattered in more than 10,000 pages, divided into hundred of papers written by a hundred different mathematicians. Since the probability of finding a mistake on a very long mathematical text is not negligible, we may legitimately asks about the necessity of such long and complex proofs and their reliability, particularly when taking into account the next turn of the screw: the famous birth of the so-called computer-assisted, or computed-based, proofs of the “four colour problem” and the solution of “Kepler´s conjecture”.
• Will computers in the future be able to formulate interesting conjectures and prove theorems?
• Are mathematicians an endangered species?
• Will the mathematics of tomorrow be full of proofs that depend upon calculations that can only me performed by very powerful computers?
• Has the existence of the modern computer changed mathematics into an experimental science?
• Will computers help mathematicians to successfully treat the more complex models of nature?
These are just a sample of the interesting questions that were addressed by the participants in the debate, giving ideas and opinions that are collected in this special issue of Philosophical Transactions, whose very clarifying and stimulating reading we strongly recommend.
A great book! It is at least as valuable as the exhibition that it catalogues. It will be appreciated by anybody who is interested in or curious about Kurt Gödel.
The life’s work of Gödel ranks among the highest from the point of view of pure science. At the same time it must be seen in the context of the intellectually productive Viennese atmosphere that was present in the first decades of the 20th century and of the following political disaster.
The catalogue is divided into three parts: Gödel's life, Gödel's work and Gödel's Vienna. It is beautifully illustrated, with photographs, documents and letters. We have never been given a closer look at the true Gödel; we see a copy of a school report of the eleven-year-old Kurt that exhibits only the best grades, with the only exception being a second best in mathematics! We also see copies of official documents concerning Gödel's PhD and his Habilitation, and we see photographs of Adele, who was seven years older than Kurt and who, according to O. Morgenstern, “saved his life”.
Between 1929 and 1937, Gödel produced his breathtaking results on the completeness of the calculus for 1st order logic (his PhD topic), on the incompleteness of formal systems (the topic of his Habilitationsschrift) and on the relative consistency of the axiom of choice and the generalized continuum hypothesis. But not only did he never get a permanent position at the university of Vienna, he was also deprived of his ‘venia legendi’ by the National Socialists. The relevant paperwork reproduced in the catalogue shows an oppressive reality. In June 1940, Gödel was again granted the title of “Dozent”; he never picked up the certificate, though, having emigrated to the US earlier that same year.
Other emigrants from Vienna included Karl Menger, O. Morgenstern, Olga Taussky, Otto Neurath and Rudolf Carnap, to mention just a few who are discussed in the catalogue.
In Princeton, Gödel focused on philosophy and the theory of general relativity. He reconstructed a formal version of the ontological proof of the existence of God, which he did not want to be published for fear that people would conclude that he really believed in God. Gödel also presented a solution to the field equations allowing time travel into the past. His letters to his mother give a detailed account of his intellectual hikes. Photographs show him chatting with Albert Einstein and enjoying himself in the garden of his house.
Some of the material as well as background information may be found at www.goedelexhibition.at/. We must congratulate the organizers on creating such an impressive exhibition and the authors on producing this catalogue. I wish there could be more books like this, which make mathematicians comprehensible in their cultural and political contexts.