Books by Karl Sigmund
We list below various books written by Karl Sigmund. We give extracts from reviews and publisher's information for these (where appropriate). The books are listed in chronological order.
- Ergodic Theory on Compact Spaces (1976), by Manfred Denker, Christian Grillenberger and Karl Sigmund.
1.1 Review by: Bill Parry.
Mathematical Reviews MR0457675 (56 #15879)
For the two decades preceding 1960 ergodic theory was a comparatively dormant subject that could be viewed (through the books of Hopf, Halmos and Jacobs for example) as a unity. Since then the subject has grown in popularity, activity and scope, making an overall view nearly impossible except perhaps in survey article form ... The most we can expect in book form are specialised personal accounts and these are appearing regularly. Such is the book under review, which reflects, to a considerable extent, the separate but overlapping interests of three authors. The subject they have chosen is restricted, in that it concerns topological ergodic theory and omits the volatile areas of Bernoulli and loose Bernoulli theory, ergodic number theory, cohomology of group actions, etc. Nevertheless, their work is a welcome addition to the growing number of books on ergodic theory. Students will find in it easy passages to a number of frontier topics and sound foundational material frequently omitted from courses and other sources. ... The wealth of information provided by this timely treatise cannot be conveyed in a short review. Suffice it to say that current ergodic theorists will find the book indispensable.
- Evolutionstheorie und dynamische Systeme (1984), by J Hofbauer and Karl Sigmund.
This book was translated into English with the title The Theory of Evolution and Dynamical Systems. See  below for reviews of the English version.
- The Theory of Evolution and Dynamical Systems (1988), by J Hofbauer and Karl Sigmund.
3.1. From the Publisher:
This textbook is an introduction to dynamical systems and its applications to evolutionary game theory, mathematical ecology, and population genetics. This first English edition is a translation from the authors' successful German edition which has already made an enormous impact on the teaching and study of mathematical biology. The book's main theme is to discuss the solution of differential equations that arise from examples in evolutionary biology. Topics covered include the Hardy-Weinberg law, the Lotka-Volterra equations for ecological models, genetic evolution, aspects of sociobiology, and mutation and recombination. There are numerous examples and exercises throughout and the reader is led up to some of the most recent developments in the field. Thus the book will make an ideal introduction to the subject for graduate students in mathematics and biology coming to the subject for the first time. Research workers in evolutionary theory will also find much of interest here in the application of powerful mathematical techniques to the subject.
3.2. Review by: Alexander Gimelfarb and Steven Orzack.
BioScience 39 (11) (1989), 820-821.
Despite its title, 'The Theory of Evolution and Dynamical Systems' is much more a unified treatment of dynamical systems arising in different areas of evolutionary biology than it is a discussion of evolutionary theory. The virtue of this focus is that it allows one to readily see the mathematical unity of models motivated by distinct biological subjects. The book is divided into two main parts. The first part, an introduction to the qualitative theory of differential and difference equations, is based entirely on examples from biology. ... The second part of the book contains a unified treatment of the dynamical systems discussed in the first part. ... What does one learn from this book? The biologists will learn of the mathematical unity of seemingly distinct biological problems, of the wealth of mathematical complexity hidden behind even relatively simple biological models, and of the mathematical rigour that can be usefully applied in the analysis of such models. The mathematician will learn of the variety of biologically motivated problems still unsolved or not fully analysed.
3.3. Review by: Thomas G Hallam.
SIAM Review 32 (2) (1990), 322-323.
Hofbauer and Sigmund indicate in the Preface that " ... crucial aspects of theoretical biology can only be captured by mathematical modelling; and just as important as the mathematical applications in biology are the biological motivations to mathematics." The maturity of the field of mathematical biology does indeed rest on the interplay between the mathematics and the biology. Any attempt to do justice to both areas requires sincere efforts; it is clear that the authors have invested much effort in this work. It is also evident that this book is written for mathematicians. Even the biologically oriented sections often include models and mathematics that many biologists would find challenging. The book addresses a spectrum of topics ranging from the prebiotic evolution of macromolecules to population genetics, dynamic population, and community ecology, and to game-theoretic modelling of animal behaviour.
3.4. Review by: H Resit Akcakaya.
The Quarterly Review of Biology 64 (4) (1989), 493.
This textbook is an introduction to the use of mathematics in four related areas of biology: population ecology, population genetics, sociobiology and prebiotic molecular evolution. The main emphasis is the analysis of differential equations with examples of their use in models of ecological and evolutionary processes. ... Although the motivation of discussions throughout the book is biological, the book will be more accessible to students of mathematics than to students of biology. It will be very useful and stimulating for mathematics students who want to shift their interest to- wards biology, and it may also be helpful for mathematically oriented students of biology.
3.5. Review by: Sabin Lessard.
American Scientist 79 (2) (1991), 180.
Those interested in dynamical systems with particular reference to evolutionary biology will appreciate this textbook. Some of the most classical results and more recent developments in population genetics, ecology, evolutionary game theory and prebiotic evolution are presented. Special attention is given to the Lotka-Volterra equations and game dynamics. Applications are found for several mathematical techniques, including the Ljapunov functions, the Poincaré maps, the Hopf bifurcations, the Shahshahani gradients, the Perron-Frobenius theory and the Poincaré Bendixson theorem. Throughout the book there are exercises for students and notes for teachers or investigators who are interested in knowing more about the subject. The treatment is clear, and the book is quite appropriate for an advanced course in differential equations and their applications.
3.6. Review by: Gabriela Schranz-Kirlinger.
Mathematical Reviews MR1071180 (91h:92019).
This book is an extended English translation of the 1984 German version (Evolutionstheorie und dynamische Systeme: Mathematische Aspekte der Selektion, 1984). The authors' goal is indicated in the introduction. "It should be (a) an introduction to the theory of dynamical systems (and in particular the qualitative theory of differential equations), based entirely on examples from biology; and (b) a survey of recent developments in four branches of the theory of evolution, namely population genetics, mathematical ecology, prebiotic evolution of macromolecules, and game theoretic modelling of animal behaviour ... " The field of mathematical biology or biomathematics rests on the interplay between mathematics and biology. The authors' position is clear, their main emphasis lies on the first; the book is written by mathematicians for mathematicians. ... The authors' hope "to point out some interesting sights along the way from undergraduate mathematics to current research'' is successfully achieved. More than that, this outstanding work clearly represents biologically motivated mathematics interesting for graduate students in mathematics and biology coming to the subject for the first time, as well as for research workers in evolutionary theory.
- Games of Life: Explorations in Ecology, Evolution, and Behaviour (1994), by Karl Sigmund.
4.1. From the Publisher:
Life is often a matter of gambles, pay-offs, and trade-offs, just like a game. This book takes readers on a tour through the games and computer simulations that are actually helping to advance knowledge in such fields as ecology, evolution, and animal behaviour. Although the book deals with questions of vital importance, like sex and survival, it does so in the lively, entertaining spirit of game-playing. It starts with artificial life and self-replicating automata, a topic ideally suited for a computer-games approach. The book goes on to study pursuit games between predators and prey, and chaotic motion and its role in ecology. Games of chance and statistical paradoxes illuminate the randomness in molecular evolution, while some bizarre double games played by chromosomes help explain the laws of population genetics. Other topics include courtship, ownership, partnership, and brinksmanship-illustrated through the game of poker and computer tournaments. No other book explains so well why scientific observations and insights can be structured as the rules of a survival game, and what happens when they are assembled on a computer or in the mind and allowed to run their course. General readers as well as professionals and students in ecological, evolutionary, and behavioural studies will find this a fascinating and informative work.
4.2. Review by: Times Higher Educational Supplement.
Karl Sigmund's 'Games of Life' is a beautifully written and, considering its relative brevity, amazingly comprehensive survey of past and current thinking in "mathematical" evolution. Just as games (at least, the human variety) are supposed to be fun, so too is 'Games of Life' - the witty section headings, the relaxed style and the clarity of the explanations make the book as enjoyable to read as a Marx Brothers film (to which there is a reference in the book) is to watch.
4.3. Review by: Peter Yodzis.
Science, New Series 264 (5156) (1994), 294-295.
There is a style of popular scientific writing that draws its narrative energy from the personalities of a few prominent scientists and the drama that flows from their obsessions. The best of this genre are well worth the attention of students and practitioners of science, but these readers are also well served by something a little meatier, in the manner of George Gamow or Erwin Schrodinger in their "popular" mode. Karl Sigmund's Games of Life is firmly in this latter tradition, though it does contain a few (quite entertaining) biographical asides. The book is a semipopular account of theoretical evolutionary biology, with an emphasis on behavioural phenomena and on game-theoretical methods. The tone is genial and playful. Although the book is about mathematical ideas, Sigmund has opted to avoid explicit mathematics (equations). Presumably this is meant to make the book more palatable to a readership of biologists, but there are a few spots in the book where an equation or two would make the argument a lot more transparent. Sigmund introduces his book with a spirited defence of the use of mathematical thinking in the context of biological problems. He reminds us, for example, that Mendel was a student far less of biology than of mathematics; and later in the book he goes so far as to suggest that Mendel's mathematical training accounts for the otherwise enigmatic circumstance that it was he and not his contemporary Darwin who laid the genetic foundation that was to support Darwin's own ideas. As one is carried along by Sigmund's persuasive account here, nothing seems more natural than to apply mathematical thinking in biology-one can almost imagine the day when a semipopular book on mathematical biology will contain a few equations. For Sigmund, mathematics is the essential tool of the thought experiment, the exploration of the explanatory power of some what if? proposition.
4.4. Review by: Ethan Akin.
The Quarterly Review of Biology 69 (4) (1994), 573-574.
Sigmund knows, and intends to show, that to identify mathematics by solving huge equations is to confuse a practice with tools used therein. Mathematics is better thought of as a style of thinking hinted at with words like "modelling" and "abstraction," which are easier to demonstrate than to describe. This book is a demonstration conducted with very great style indeed. Sigmund's initial description is a bold admission inasmuch as love of mathematics is a perversion sufficiently rare as to render the phrase "popular mathematics book" oxymoronic. His confrontation with the tastes of his prospective readers is deliberate in that he intends to defend as well as to illustrate the use of mathematics in biology. His first chapter provides an explicit argument for the patterns of thinking, which his later chapters illustrate. Sigmund argues for the value of thought experiments. These are usually hypothetical models that are somewhat removed from actual data. It is this a priori character that requires a defence, especially in light of the imperial arrogance shown by some mathematicians as they sail off to colonise neighbouring subjects. For example, Rene Thorn, in the delightful tradition of Cartesian rationalism, suggested that embryologists abandon the problem of morphogenesis to the mathematicians who would, no doubt, solve it in short order.
4.5. Review by: Jeffrey R Lucas.
Ecology 75 (8) (1994), 2468-2469.
There are a number of writers, including Richard Dawkins, Steven Vogel, and Stephen Gould, whose writing style makes their work worth reading just for the prose. Karl Sigmund is another name to add to the list. Sigmund's Games of Life is loosely centred on a fairly eclectic range of biological games, with an intended audience of "potential or actual students and the interested layperson." The book covers inherently mathematical themes but offers only the logic behind the models and their predictions, without any of the math, hence the accessibility to interested laypersons. Sigmund's lucid style and use of historical anecdotes make this book eminently readable. The book doesn't break any new ground, but it is a wonderful introduction to the logic of some of the models that have been developed in evolutionary ecology.
- Evolutionary Games and Population Dynamics (1998), by J Hofbauer and Karl Sigmund.
5.1. From the Publisher:
Every form of behaviour is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centred not on the concept of rational players but on the population dynamics of behavioural programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms.
5.2. Review by: Susan Holmes.
Journal of the American Statistical Association 95 (450) (2000), 688.
This book is written by well-known specialists of dynamical systems and their applications to ecology. This presentation takes game theory from Von Neumann's initial setting through John Nash's work on equilibrium seen as a branch of dynamical systems, and explains the applications of game theory to biology. Evolutionary Games and Population Dynamics is definitely a book that requires some mathematical training, and biologists desiring an introduction to the subject would benefit from a more concrete, hands-on book .... Students in applied mathematics, however, will find Evolutionary Games and Population Dynamics exactly along their lines, with just enough applications given to make the equations come to life.
5.3. Review by: Steven D Carroll.
The Quarterly Review of Biology 74 (3) (1999), 347.
In essence, this is a mathematical textbook, the main subjects of which are replicator dynamics and Lotka-Volterra equations. The book is divided into four parts: Dynamical Systems and Lotka-Volterra Equations, Game Dynamics and Replicator Equations, Permanence and Stability, and Population Genetics and Game Dynamics. Each part contains an exhaustive compilation of mathematical theorems, many of which have been added to the literature within the last decade. The book lacks (by design ) extensive biological discussion, so interpretation of these results is generally left to readers. ... Relatively complicated theorems and proofs comprise a large portion of the book, and it is therefore not recommended for those who are not mathematically inclined. In fact, many theorems are stated without proof, and in stead are given as exercises for readers to complete. For the biologist who is mathematically inclined or the mathematician interested in biology, however, this volume is rich in results and likely to provoke stimulating thought.
5.4. Review by: Gabriela Schranz-Kirlinger.
Mathematical Reviews MR1635735 (99h:92027).
The book under review is a very nice further development of its ten-year-old predecessor (The theory of evolution and dynamical systems, 1988) by the authors. Not only has the title been modified, but also the contents have been thoroughly reworked and thus adapted to today's topics of interest in the field of biomathematics. The book is totally restructured and contains much new material, mainly in game theory, especially in its evolutionary and dynamical aspects. Game theory is approached in terms of dynamical systems. ... Summarizing, this book is written in the well-known authors' usual clear, elegant and motivating style.
- Kurt Gödel - The Album (2006), by J Dawson, K Mühlberger and Karl Sigmund.
6.1. From the Publisher:
Time Magazine ranked him among the hundred most important persons of the twentieth century. Harvard University made him an honorary doctor "for the discovery of the most significant mathematical truth of the century". He is generally viewed as the greatest logician since Aristotle. His friend Einstein liked to say that he only went to the Institute to have the privilege of walking back home with Kurt Gödel. And John von Neumann, one of the fathers of the computer, wrote: "Indeed Gödel is absolutely irreplaceable. He is the only mathematician about whom I dare make this assertion." This book wants to give a simple, intuitive and easily digestible introduction to Gödel's life and work, meant for readers interested in the human and cultural aspects of science. Its starting point was the preparations for an exhibition on Kurt Gödel, on the occasion of his hundredth birthday. An exhibition has something of a walk in it, and that's just what we want to offer: a walk with Gödel. Albert Einstein enjoyed such walks very much. So one can enjoy Gödel.
6.2. Review by: Jeremy Gray.
Mathematical Reviews MR2242887 (2007d:01010)
This attractive book, in English and German, is an account of the life of Kurt Gödel in words and pictures. It grew out of preparations for an exhibition on Gödel as part of the commemorations of the 100th anniversary of his birth, and extracts from it have appeared in the April 2006 issue of the Notices of the American Mathematical Society. The authors offer it to readers "interested in the human and cultural aspects of science''. They trace Gödel's life from birth, through school and university, the first of his great discoveries, the highly stressful time of his emigration to Princeton in 1939, and his later work, to his death from inanition in 1978. They show the great impact his work had, document his close friendship with Einstein, and devote a chapter to Gödel's Vienna and the Vienna Circle. In an appendix they provide Gödel's own brief summary of his incompleteness theorem and a longer account by Menger, rightly judging that this is not the occasion for a more detailed account. They provide a rich introduction to the study of Gödel's ideas that will more than answer the question of who Gödel was, that will go some way to answering the question of what he did, and that will stimulate young and not so young readers to find out more.
6.3. Review by: Ralf Schindler.
European Mathematical Society.
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".
6.4. Review by: Fernando Q Gouvêa.
Mathematical Association of America.
2006 is Kurt Gödel's centenary year, and this book is a worthy way to celebrate. Based on the catalog for a Gödel exhibit, this "album" contains a wealth of photographs and documents that illustrate the life and ideas of one of the most important mathematicians of the twentieth century. As the title indicates, this is an album: a collection of images, of people, places, and texts, with captions in German and English. There are all sorts of neat things here. Many photographs of Gödel are included, one of which is described as a rare image of the man without his glasses. On page 19, there is a high school report card, in which the grades are the highest possible ("sehr gut") on every subject but one. The one exception, of course, is mathematics, in which his grade is a mere "gut". There are many photos of Gödel's contemporaries, including many of the members of the "Vienna Circle." There is even a note, by the director of the Institute for Advanced Study, in which he decides that the speakers at Gödel's funeral should discuss his work on set theory and on logic, but not his "minor" contribution to general relativity.
6.5. Review by: Wilfried Sieg.
History and Philosophy of Logic 29 (1) (2008), 94-96.
The year 2006 is the centenary of Gödel's birth. The preparation of 'Gödels Jahrhundert', an exhibition shown in Vienna from 11 July to 8 August 2006, was also the starting-point for this book. The book's subtitle 'Das Album - The Album' hints at the richness of the photographic material that forms its backbone: early family photos in Brno, photos taken during Gödel's days in Vienna and Princeton, and reproductions of many fascinating documents. The visual material is surrounded by text in English and German; the text explains its significance and draws frequently illuminating connections. The book is divided into three main parts that follow a Preface by the German poet Hans Magnus Enzensberger and an Introduction; the latter ends with the remarks: 'As he [Gödel] told Hao Wang in one of their long interviews: ''I do not fit into this century''. And yet he left his mark on it, maybe precisely because he remained a stranger.' Perhaps the book's strongest point is that it represents Gödel's strained relationship with his century and illuminates the 'maybe precisely because'. It does so by showing an illustrated history of Gödel's life in the first part, entitled 'Gödels Leben - Go
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