Book Description

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik–Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants. ... Read more

Book Description

When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied.

The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors."

This mathematical masterpiece will appeal to all mathematicians working in algebraic topology. ... Read more

Reviews (2)

A landmark, that's it.
As the title of the series suggests, this is another "landmark" in books of mathematics. In fact, this was the only book on homological algebra for a certain period of time. I don't think this is a fungus, but with other modern good textbooks available (Weibel, for example) I am not inclined to give it "five stars".

H. Cartan and S. Eilenberg are the authors
I believe this is volume #19 of the Princeton Math. Ser. entitled Homological Algebra. The authors are H. Cartan and S. Eilenberg. H. Cartan is one of E. Cartan's sons. ... Read more

Book Description

From the reviews:"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature...(S.Y. Husseini in Mathematical Reviews, 1976) ... Read more

Reviews (1)

This might take a while...
The earlier chapters are quite good; however, some ofthe advanced topics in this book are better approached (appreciated) after one has learned about them elsewhere,at a more leisurely pace.For instance, this isn't the best place to first read about characteristic classes and topological K theory (I would recommend, without much hesitation, the books by Atiyah and Milnor & Stasheff, instead).Much to my disappointment, the chapter on spectral sequences is quite convoluted. Parts of 'user's guide' by Mclearywould certainlycome in handy here (which sets the stage rather nicely for applications).

So it turns out that supplemental reading (exluding Whitehead's massive treatise) is necessary to achieve a better understanding of algebraic topology at the level of this book.The homotopical view therein will be matched (possibly superseded) by Aguilar's book (forthcoming, to which I am very much looking forward).

Good luck! ... Read more

Book Description

The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus. ... Read more

Reviews (4)

A complete book by very erudite authors
I actually read this entire book--it is quite long and dense. Actually I took the course from the author Jerry Marsden at Caltech and Tutor (Jerry's friend and co-author) gave a guest lecture while visiting. We flew through the entire thing and ch 9 on lie groups of his mechanics and symmetry text in a short 10 weeks! My background in math was relatively weak when taking the course so it was a little hard to keep up; i.e. I came from an engineering background. Anyway, it is probably the most complete/diverse text I've come across on the subject. Of course, it's actually more of a monograph than a text. Since I've read the whole thing, I have to admit there are "several" typos. But as it is that most people can't even write a damn email without a typo or two, the book really does a good job considering it is 800 pages of mostly dense mathematical rigor. I imagine that if I wrote 800 pages of mathematical symbols in latex, that I might forget a tilde or put something as subscript that should have been superscript here or there! None of these errors really matter too much-they should not hinder one's understanding. All and all I think that this book is a great ref, although I've never seen the index, if one exists. For the beginner, also check out Boothby's book, which covers a lot of the same material but tones it down a bit.

A Unique Reference
Students of mathematical physics in general, and general relativity in particular, face a formidable challenge in attempting to find coherent, readable references on manifold theory and tensor analysis.I think it fair to say that for every well-written work on the subject, there are ten that do more damage than good.Very few texts can claim to (1) be clear enough to assist the person who is studying alone, (2)offer valuable PHYSICAL insight into the subject, and (3) pass the standards of rigor that mathematicians would impose.Abraham, Marsden, and Ratiu manage to accomplish all three of these goals in this profoundly useful text.I studied from the first edition and I have taught from the second.The two chapters on differential forms, Hodge star duality, integration on manifolds, and the generalized Stokes' Theorem alone are worth the price of the entire book.I am unaware of any other reference which which treats differential forms with the same combination of clarity, physical motivation, and mathematical rigor.The concluding chapter on applications offers one of the clearest introductions to the relativistic form of Maxwell's equations to be found in any text.For students of physics who want to see the mathematics "done right," one would be hard pressed to do better than Abraham, Marsden, and Ratiu.

mixed bag: many virtues but many weaknesses
I took a course taught by the 3rd author (Tudor Ratiu) at UCSC using this book; I found both good and bad in it.Much of the bad for me was overcome by the inspiring and energetic presentation by one of the authors.One mayview this book as basically a detailed elaboration of the"preliminary" chapters of the book "Foundations ofMechanics" by the 1st 2 authors. The strengths of this book are (a)the treatment which is general enough to include infinite-dimensionalmanifolds and not just the finite-dimensional case (most books just talkabout the finite-dim'l case) and (b) the attempt to cover all theorems"full strength" (in the greatest generality obtaining thestrongest conclusions from the weakest hypotheses). Neither of these (notcounting the many typos) recommends this as a first or even second text forstudents, but it's hard to find any other books that treat the material atthe same level of generality and precision, which is a must if attempting"hard" global analysis in areas such as fluid mechanics (from ageometric point of view).Correction of the many typos could make this anindispensable reference book for those requiring the techniques discussed.More discussion of finite-dimensional examples before jumping toinfinite-dimensional ones (e.g. discussing finite-dimensional Grassmanniansbefore jumping to the infinite-dimensional Banach manifold version) couldmake this into a tolerable text.

As it is, it's problematic, aggravating,and not for the faint of heart, but not without its virtues.

Possiblealternatives for the infinite-dimensional point of view are Lang'smanifolds book or some volume of the expensive multi-volume treatise onanalysis by Dieudonne.

Poorly writen, filled with errors, very long, poorly indexed
We used this book in a graduate course at UCLA. The professor had to hand out a list of all the errors we encountered, and it was about ten pages typewritten. The professor, Geoff Mess, wrote at the top of this list thatmany of the students had complained about this book, and that it was adisappointment to him as well.I often found myself scanning hundreds ofpages in search of what should have been contained in their sparse index.The book is unnecessarily long and wordy for the matter covered. In theintroduction, the authors mention that they invite comments from thereaders. It seems that they depend on their readers to correct theircopious errors and their poor writing. ... Read more

Book Description

Fundamental topological facts, together with detailed explanations of the necessary technical apparatus, constitute this clearly written, well-organized three-part text. Part 1 deals with certain classic problems without using the formal techniques of homology theory; parts 2 and 3 focus on the central concept of combinatorial topology, the Betti groups. Numerous detailed examples.
... Read more

Book Description

Theorems of Leray-Schauder Type and Applications presents a systematic and unified treatment of Leray-Schaudercontinuation theorems in nonlinear analysis. In particular fixed theory is established for many classes of maps, for example, contractive, non-expansive, accretive and compact. This book also presents concidence and multiplicty results. Many applications of current interest in the theory of nonlinear differential equations are given to complement the theory. The text is essentially self-contained so it may also be used as an introduction to topological methods in nonlinear analysis. ... Read more

Book Description

Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe non-abelian homological algebra and to address homotopy-theoretical issues in a variety of fields, including algebraic K-theory. This book supplies a modern exposition of these ideas, emphasizing model category theoretical techniques.

Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, Postnikov towers, and bisimplicial sets. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. Interspersed throughout are many results and ideas well-known to experts, but uncollected in the literature.

Intended for second-year graduate students and beyond, this book introduces many of the basic tools of modern homotopy theory. An extensive background in topology is not assumed.

Series: Progress in Mathematics, Vol. 174 ... Read more

Reviews (1)

Excellent book!
Gives a well-written and concise treatment of developments in an area of topology that has seen considerable progress in the past 50 years. The only other general expository books in this area are more than 20 years old. This is particularly important because the book unifies many seemingly disparate results and approaches. Even classic constructions (like the Hurewicz homomorphism) are give modern (and very concise) interpretations. I strongly recommend this book to students and researchers in algebraic topology. ... Read more

Book Description

Concise, rigorous introduction to homology theory features applications to dimension theory and fixed-point theorems. Lucid coverage of the field includes examinations of complexes and their Betti groups, invariance of the Betti groups, and continuous mappings and fixed points. Proofs are presented in a complete and careful manner. A beneficial text for a graduate-level course, "this little book is an extremely valuable addition to the literature of algebraic topology."—The Mathematical Gazette. Unabr. repub. of 1952 ed.
... Read more

Reviews (3)

readable
proofs are very detailed almost pedagogic. the author always points out the goal first then digs in. the only flaw is its visual presentation - rather condensed type set. i think Pontryagin was a truly educational teacher. good for self study.

A cheap classic by one of the masters
This book covers, as the title says, the foundations of combinatorial topology. What that refers to, essentially, is the topology of polyhedra and the machinery of simplicial homology groups. The first part of the book establishes the basic facts about these topological spaces and about abstract complexes as well. The second part shows the topological invariance of simplicial homology. The third features applications of the material to fixed point theory.

Throughout the book there is also a recurring digression on dimension theory, which culminates in a proof of the very non-trivial fact that manifolds have the appropriate topological dimension.

The only formal pre-requisites for this book are basic linear algebra and point-set topology, as covered in any advanced calculus or elementary real analysis course.

The presentation is very concise and lucid throughout. I can imagine some people not liking such an extremely concise style, but the whole book is so logical that it works very well.

A great little book
This is a beautiful little book on a very pretty subject. Lev here develops very comprehensively the basic properties of polyhedra, as well as many of their basic applications. The first chapter is spent on basic topological properties of these spaces, with some applications to dimension, very explicitly set aside from the main development. The second chapter focuses on proving the topological invariance of the Betti groups, with applications of the techniques used-again carefully set aside-to dimension theory and Brouwer's fixed point theorem. The third chapter, titled Continuous Mappings and Fixed points, has as a goal proving a simple fixed point theorem. Some simple constructions and concepts of algebraic topology, such as homotopy and the cylinder construction, are introduced along the way. Unfortunately, not too long after the publication of this book, the author began to devote his energies to applied mathematics and preventing Jews from polluting Russian mathematics-but don't let that influence your possible purchase of this book. ... Read more

Book Description

This elegant little book discusses a famous problem that helped to define the field now known as topology: What is the minimum number of colors required to print a map such that no two adjoining countries have the same color, no matter how convoluted their boundaries. Many famous mathematicians have worked on the problem, but the proof eluded fomulation until the 1950s, when it was finally cracked with a brute-force approach using a computer. The book begins by discussing the history of the problem, and then goes into the mathematics, both pleasantly enough that anyone with an elementary knowledge of geometry can follow it, and still with enough rigor that a mathematician can also read it with pleasure. The authors discuss the mathematics as well as the philosophical debate that ensued when the proof was announced: Just what is a mathematical proof, if it takes a computer to provide one -- and is such a thing a proof at all? ... Read more

Book Description

Following their introduction in the early 1980s, o-minimal structures have provided an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. This book gives a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. It starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists. ... Read more

Book Description

Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem.The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings.The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field. ... Read more

Book Description

This book develops some of the extraordinary richness, beauty, andpower of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a livingfeeling for the subject. There are many figures, examples, and exercises of varying difficulty. ... Read more

Reviews (2)

A refreshing style of writing
Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear. Thurston's book is a bit like learning by ear.

fun and geometric-intuition-minded
A must for anyone entering the field of three-dimensional topology and geometry. Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it is readable to undergraduates. Its target audience, though, is beginning graduate students in mathematics. If not already familiar with hyperbolic geometry, you might want to get an introduction to the subject first. Once with this background, though, you will discover there is another level of understanding of hyperbolic space you never realized was possible. One imagines Thurston able to skateboard around hyperbolic space with the kind of geometric understanding he conveys here.

What made Thurston so famous and successful as a pioneer in 3-d topology and geometry was his other-worldly geometric intuition. This book takes the reader along the first step of the 10000 miles of getting to that intuition. ... Read more

Book Description

This self-contained book by a leading topologist is devoted to algorithmic low-dimensional topology, a branch of mathematics that has recently been undergoing an intense development. The book contains plenty of important fundamental material, which is carefully presented. The book also contains some of the author's own original contributions. For the first time ever, it gives a full exposition of the complexity theory of 3-manifolds and a complete proof of the solution of the homeomorphism problem for Haken manifolds. The style of the book is very lively, with a lot of useful pictures, making the book enjoyable for those who like visual topology. The writing is clear and the proofs are careful and detailed. This book fills a gap in the exisiting literature and will become a standard reference for this aspect of 3-dimensional topology both for graduate students and researchers. ... Read more

Book Description

A new foundation of Topology, summarized under the nameConvenient Topology, is considered such that several deficiencies oftopological and uniform spaces are remedied. This does not mean thatthese spaces are superfluous. It means exactly that a better frameworkfor handling problems of a topological nature is used. In this settingsemiuniform convergence spaces play an essential role. They includenot only convergence structures such as topological structures andlimit space structures, but also uniform convergence structures suchas uniform structures and uniform limit space structures, and they aresuitable for studying continuity, Cauchy continuity and uniformcontinuity as well as convergence structures in function spaces, e.g.simple convergence, continuous convergence and uniform convergence.Various interesting results are presented which cannot be obtained byusing topological or uniform spaces in the usual context. The text isself-contained with the exception of the last chapter, where theintuitive concept of nearness is incorporated in Convenient Topology(there exist already excellent expositions on nearness spaces).Audience: Students and mathematicians having some basicknowledge of topological spaces and seeking a deeper insight intopology on a convenient level will find this a valuable text. It isalso useful for researchers in the field of topology and itsapplications being interested in a further development of thepresented theory. ... Read more

Book Description

This book describes fundamental groups and their topological soulmates, the covering spaces. An accomplished example of the algebraic topologist's dream come true, covering spaces are a geometric (that is, topological) structure that is completely characterized by its algebraic counterpart.

The author provides several illustrative examples that touch upon different areas of mathematics, but in keeping with the book's introductory aim, they are all quite elementary. Basic concepts are clearly defined, proofs are complete, and no results from the exercises are assumed in the text. ... Read more

Reviews (4)

Read the Adams book instead
If you just plan to skim the text and do not intend to try applying the ideas presented to actual knots, then you may not notice this small book's many errors. But if you wish to verify what the text says and try your hand at some knot calculations, then this is not the book for you. Perhaps the worst example is the author's comment that the figure-eight knot and the trefoil not have the same Conway polynomial. They don't. After an hour of calculating and recalculating, it is frustrating to discover that the author, not the reader, is the one in error. That kind of elementary error makes one question the author's basic competence and knowledge of the field.

Another error is made when giving an example of calculating the Conway polynomial for a link with two separate circles (page 68): the right-hand side of the equation should have no term in x. Figure 2.15 (algebraic representation of a braid) also has an error: the upper-right-hand braid elementary braid is b2, not b1. (The text below the diagram is correct, but the diagram itself has it wrong.)

For a beginner who is learning the subject, the necessity of sorting out the author's errors is unacceptable. A book with so many errors should have an errata (list of corrections) on the web, but I searched and found none.

I though the braid chapter was well-written. I have not studied braids before and it made the situation pretty clear.

On the plus side, the drawings are excellent, the best I have seen in any knot book. For example, figure 3.3 (page 40) has a nice diagram clearly showing various "problems" that might happen momentarily during Reidemeister moves. In this case, a picture is worth a thousand words.

I did not enjoy the author's mini-digressions into non-mathematical applications of knots. They went on too long and didn't relate well to the mathematics in the book.

Finally, this author seems to have a bit of an attitude. He makes it sound like he almost beat Kaufmann to discovering Kaufmann's bracket. Then he goes on to point out that the Celtic people discovered a form of it centuries ago (beating Kaufmann). Sounds like sour grapes to me. He makes frequent comments such as "the attentive reader will notice," which I found annoying after a while. Readers do not like to be insulted.

After a full day with this book, I am tossing it into the trash. The Knot Book by Colin Adams is solid on the math and a better overall introduction to the math side.

It is not that bad, but full of mistakes
I actually read the French version, and skimmed through the Englih one. When I read it in French, I was baffled by the number of mistakes per page. So I reread it, keeping a list of mathematical mistakes and typos(?). It averaged 1.7/page. I send it in to the French editor, but I realized that they kept the mistakes in the English version!

On the other hand, I thought explanations were pretty good.

So I would certainly not recommend it as a starter, but if you know enough of knot theory, the mistakes should keep you entertained...

This book is bad!
Don't buy this book if you're a mathematician!

Either something really disturbing has happened during one of the translations (russian->french->english), or I seriously doubt mr. Sossinsky's ability to teach anyone about knot theory.

Almost every single calculation in the book is wrong. Some of the errors are plain typo's, admitted. But others are so disturbingly wrong that I had to read the passages several times to believe that a mathematician could have written this.

One notable example is when the author calculates (correctly for once) the Conway polynomial of the trefoil knot to be 1+x^2. Then goes on (this is so good, I just have to quote it):

"A calculation similar to this one shows that the Conway polynomial for the figure eight knot (Figure 1.2) is equal to x^2+1: it is the same as that for the trefoil. The Conway polynomial does not distinguish the trefoil from the figure eight knot; it is not refined enough for that."

In fact, the figure eight knot has Conway polynomial 1-x^2. Scary that an expert on knot theory can make this error (three times in a row!). -Afterall, the simplest counterexample to whether the Conway polynomial is a perfect invariant is a very, very basic thing to know!

Other mistakes are rather amusing (even whilst still being annoying). For instance, the author confuses a figure-eight knot with an unknot, shortly after casually mentioning that his intuition of space is "fairly well developed".

Another thing that annoys me as a mathematician is the author's "personal digressions", trying to explain how the minds of mathematicians work and why mathematics can be beautiful in the same way as arts and music. The worst one of them is concerned with how the author *almost* discovered the Kaufmann construction of the Jones Polynomial before Kaufmann did. (At least, that's how it sounds to me.) In my opinion, either you try to explain some math, or you do pocket philosophy. -Not both at once!

On the good side, the actual subjects treated in the book are very well chosen. (Except, the author promises twice to get back to telling about the Alexander polynomial but he never does...) (And that last thing reminds me: The book has no index!!!)

So, my advise is: read the contents pages and go learn the theory from elsewhere.

Untangling Mathematically
It is always surprising and pleasing to find that mathematicians are busy in their ivory towers looking at non-numerical concepts and even using small subjects to turn out tomes that are impenetrable to us non-mathematicians. If you want to spend a little time learning how mathematicians think about the lowly subject of knots, there is now a little book with good illustrations and explanations that may go over the heads of most people, but nonetheless demonstrates the high degree of effort in this mathematical field. _Knots: Mathematics with a Twist_ (Harvard) by Alexei Sossinsky (who is a professor of mathematics at the University of Moscow; this work is translated by Giselle Weiss) demonstrates well the complexity of a field that might at first seem unpromising but actually has important relevance to the real world.

The diagrams here, and there are many of them, are a great help. You could make your knot cross over and under an infinite number of different ways. But how different, and how can you tell the difference between one knot and another? There is, according to Sossinsky, no algorithm that works in every case of classification, not even an algorithm that can be taught to a computer. This is true even though the attempts at classification, with graphic or symbolic notation which cannot be reproduced here, are quite complicated. So, being able to tell one knot from another is the as yet unattained Holy Grail of knot theory. Interestingly, if you tie a knot, however simple, into a string, you cannot tie another knot, however complicated, into the string so that one knot will, when it meets the other, untie the string. The proof of the impossibility of one knot canceling out another is nicely sketched here. The chapters here are written more-or-less independently of one another, so that if one stumps you, you can try the next with a clean slate. For needed relief, Sossinsky has put in digressions (and labeled some of them as such) which any reader ought to be able to enjoy, like the one about the slime eel that knots itself for defenses (left trefoil knot). Some of the coincidences between knots, algebra, quantum theory, and other disparate lines of thought are really quite lovely, and indicate once again that no one knows where research in pure mathematics may lead or how practical it may turn out to be.

Sossinsky has a witty style, and acknowledges how strange this mathematical world must be for visitors. At one point in demonstrating the procedure for composing a knot from primes, he parenthetically says of the task of making a rigorous definition of what he has described intuitively, "I will leave to the reader already corrupted by the study of mathematics the task." He is a genial guide to a strange land. ... Read more

Book Description

This well-known booklet, now in its third, expanded edition, provides an informal survey of applications of singularity theory in a wide range of areas. Although the first few chapters touch briefly (and critically) on the Thom-Zeeman catastrophe theory, most of the book is concerned with more recent and less controversial aspects, covering such topics as: bifurcations and stability loss, wavefront propagation, the distribution of matter in the universe, optimization and control problems, visible contours, bypassing an obstacle, symplectic and contact geometry, complex singularities, and the surprising connections between singularities and widely disparate mathematical objects such as regular polyhedra and reflection groups. Readers familiar with the previous editions will find much that is new. Results have been brought up to date, and among the new or expanded topics discussed are delayed loss of stability, cascades of period doublings and triplings, shock waves, implicit differential equations and folded singularities, interior scattering, and more. Three new sections give an overview of the history of singularity theory and its applications from Leonardo da Vinci to modern times, a discussion of perestroika in terms of the theory of metamorphoses, and a list of 93 problems touching on most of the subject matter in the book. The text is enhanced by fifteen new drawings (there are now 87 in all) and improvements to old ones. The already extensive literature list has been updated and expanded. As a result, the book has been enlarged by almost a third. Arnol'd's goal with this edition remains the same: to explain the essence of the results and applications to readers having a minimal mathematical background. All that he asks, is that the reader have an inquiring mind. ... Read more

Reviews (2)

Outstanding introduction for life and behavioral scientists.
Catastrophe theory was the first mathematical "advance" to receive extensive coverage in the popular press since quantum mechanics. The interest and adulation afforded to chaos theory and nonlinear dynamicalsystems research barely immitated the enthusiasm and later scorn generatedby Thom and his followers. In retrospect it is hard to see what all theexcitement was about. Perhaps that's exactly why this book deserves seriousattention.

While the doctoral trained mathematician may find more faultshere than the rest of us, this book provides sufficient complexity for theprofessional, and at times is "gentle" enough for thenonmathematician. It will probably find its most useful audience amongpeople with "semi-mathematical" trainings, scientists in life andbehavioral or social sciences. For these people, who may just want to findout why Thom's initial theorems generated so much excitement andcontroversey, this book will be a readable delight.

This book should bemandatory reading for anyone claiming an interest in chaos theory but whodoes not understand how some of the overzealousness of a new paradigm canhave devestating consequences. Needless to say, those unfamiliar withcontemporary mathematical advances and behavioral/life/physicalapplications in European literarature will find this book invaluable.

Outstanding introduction for life and behavioral scientists.
Catastrophe theory was the first mathematical "advance" to receive extensive coverage in the popular press since quantum mechanics. The interest and adulation afforded to chaos theory and nonlinear dynamicalsystems research barely immitated the enthusiasm and later scorn generatedby Thom and his followers. In retrospect it is hard to see what all theexcitement was about. Perhaps that's exactly why this book deserves seriousattention.

While the doctoral trained mathematician may find more faultshere than the rest of us, this book provides sufficient complexity for theprofessional, and at times is "gentle" enough for thenonmathematician. It will probably find its most useful audience amongpeople with "semi-mathematical" trainings, scientists in life andbehavioral or social sciences. For these people, who may just want to findout why Thom's initial theorems generated so much excitement andcontroversey, this book will be a readable delight. ... Read more