Book Description

The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the text, no knowledge about category theory is expected from the reader. This book is intended for advanced undergraduates and graduate students with a basic knowledge of point set topology as well as group theory and can be used in a two semester course.

Marcelo Aguilar and Carlos Prieto are Professors at the Instituto de Matemticas, Universidad Nacional Autonoma de Mexico, and Samuel Gitler is a member of El Colegio Nacional and professor at the Centro de Investigacion y Estudios Avanzados del IPN. ... Read more

Book Description

This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other. ... Read more

Reviews (9)

fantastic introduction to general topology
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.

Didactic perfection
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.

The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.

After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author. Then, the author takes the reader to a higher level of abstraction, wherein he asks the reader to consider all of the continuous functions on a metric space, and turn this collection into a metric space of a special type called a normed linear space, and, more specifically, a Banach space. Thus the author introduces the reader to the field of functional analysis.

A lengthy introduction to topological spaces follows in chapter 3. The author motivates well the idea of an open set, and shows that one could just as easily use closed sets as the fundamental concept in topology. And, most important for functional analysis, he introduces the weak topology, and shows how to obtain the weakest topology for a collection of mappings from a topological space to a collection of other topological spaces. The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.

Compactness, so essential in all areas of mathematics that make use of topology, is discussed in chapter 4. It is motivated by an abstraction of the Heine-Borel theorem from elementary real analysis, and the author shows how well-behaved things are on compact topological spaces. Some important theorems are proved in this chapter, namely Tychonoff's theorem, the Lebesgue covering lemma, and Ascoli's theorem.

Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces. Chapter 5 is an in-depth discussion of separation, and the reader again confronts function spaces, and their ability (or non-ability) to separate the points of a topological space. Spaces that allow such separation to occur are called completely regular, and this property has far-reaching consequences in analysis and other areas of mathematics. The Stone-Cech compactification is discussed as an imbedding theorem for completely regular spaces, analogous to one for normal spaces.

The intuitive idea of a space being connected is given rigorous treatment in chapter 6. Certain pathologies can of course arise when discussing connectedness, and the author shows this by discussing totally disconnected spaces, remarking that such spaces are very important in dimension theory and representation theory. Indeed, computational and fractal geometry is much harder to study because of the existence of these spaces.

Chapter 7 is important to all working in numerical analysis, wherein the author discusses approximation theory. The Weierstrass approximation and the Stone-Weierstrass theorems are discussed in detail.

A slight detour through algebra is given in chapter 8. Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.

Banach spaces make their appearance in chapter 9, with the three pillars of the theory proven: the Hahn-Banach, the open mapping, and the uniform boundedness theorems. These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces. The theory of Banach spaces is very extensive, but this chapter gives a peek at this very interesting area of mathematics.

Banach spaces with an inner product are considered in chapter 10. These of course are the familiar Hilbert spaces, so important in physics and the subject of a huge amount of research in mathematics. The presence of the inner product allows constructions familiar from ordinary finite-dimensional vector spaces to carry over to the inifinite-dimensional setting, one example being the transpose of a matrix, which is replaced in the Hilbert space setting by a self-adjoint operator.

As a warm-up to the infinite-dimensional theory, finite-dimensional spectral theory is considered in chapter 11. The famous spectral theorem is proven. Then in chapter 12, the reader enters the world of "soft" analysis, wherein topological and algebraic constructions are used to study linear operators on spaces of infinite dimensions. Putting an algebraic structure on a Banach space gives a Banach algebra, and then the trick is deal with the spectrum of an element of this algebra. The reader can see the interplay between algebra, topology, and analysis in this chapter and the next one on commutative Banach algebras. Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.

Good Classical Introduction to Banach Algebras
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.

I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled 'General Topology',as well as Hocking and Young's old 'Topology'. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators -- and by implication, the Spectral theorem -- need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.

These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate.

Topology Classic
This book was recommended for our analysis course (final year at Adelaide University). It helped me pass the course but more importantly, gave me an interest in metric spaces and topology. The book is an excellent communicator and nearly 20 years after I have read it I am looking out for a secondhand copy!

One of the best written Mathematics Books ...
I became aquainted with this book many years ago and I still read it ... and send students off to read it. The book is written by an incredible expositor who was and still may be at Colorado College in Colorado. It is always the book that first comes to mind when someone asks for a reference on any of the subjects it covers. These include point set topology, analysis (Not including integration or measure theory), and operator theory. It is introductory. This merely makes you wish the author would have written several advanced sequels to this amazing book.

This book has my highest recommendation. Every mathematics student should own a copy ... ... Read more

Book Description

With a nice balance of mathematical precision and accessibility, this text provides a broad introduction to the field of topology. Author Sue Goodman piques student curiosity and interest without losing necessary rigor so that they can appreciate the beauty and fun of mathematics. The text demonstrates that mathematics is an active and ever-changing field with many problems still unsolved, and students will see how the various areas of mathematicsalgebra, combinatorics, geometry, calculus, and differential equationsinteract with topology. Students learn some of the major ideas and results in the field, do explorations and fairly elementary proofs, and become aware of some recent questions. By presenting a wide range of topics, exercises, and examples, Goodman creates an interactive and enjoyable atmosphere in which to learn topology. ... Read more

Book Description

Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. ... Read more

Reviews (5)

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Good attempt
When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the concepts of differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

Covers a lot of ground . . . but not always well
Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists. I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered. However, if a proof actually is presented, I expect this proof to be correct. In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

flawed and incomplete
Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else.

Great introduction to mathematical physics.
This book is written by physicists. Like a book by M. Nakahara it describes basics of diff geometry and topology. Though it stresses physical intuition more than formal definitions. I especially liked discussion of fiber bundles and characteristic classes. Highly recommended. ... Read more

Book Description

This elegant book by distinguished mathematician John Milnor, provides a clear and succinctintroduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. ... Read more

Reviews (5)

Take full advantage of the clear, encompassing exposition:
Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.

This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).

The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply.

Compact, readable text on the topology of manifolds
This book is very short, which is quite an asset for a math book to have. It also requires little knowledge of math beyond advanced calculus and point-set topology. I found it extremely readable, and I greatly enjoyed it. I recommend it highly, and especially enjoyed the proof on page 8 of the fundamental theorem of algebra. How far we've come since Gauss' first proof...

A good place to start
This book is exceptionally well written and easy to read. Milnor proves a major result on almost every page. One learns a lot per unit time spent on this book. Despite being less than 80 pages, the book covers a significant amount of material in a clear concise manner.

Excellent starting point for differential topology
One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.

An excellent introduction to topology of manifolds.
Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book. ... Read more

Book Description

The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature.

Key features:

* Basic results presented in an accessible way for both pure and applied mathematicians

* Extensive exercises make the work suitable as a textbook for use in graduate courses

* Full proofs included in introductory chapters; only basic knowledge of functional analysis required

* Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series

* Selected research topics presented with recommendations for more advanced topics and further reading

* Open problems to stimulate further research

"An Introduction to Frames and Riesz Bases" will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference. ... Read more

Reviews (1)

A new tool!
The subject of the book is a new tool in math, with a host of exciting applications. Of course, the subject has roots in classical ideas from harmonic analysis. But the book covers an explosive and exciting variety of developments since roughly 1990, and it is presented in the form of a graduate text. The basic idea begins with linear algebra, and progresses to expansions in function spaces, and multiresolutions. It will be useful to anyone who wants to learn from scratch about the underlying principles, the new results, and the applications. It is well written. A student of mine picked it up accidentally from my desk, and couldn't put it down. After awhile, he had completely forgotten what he came to see me about. It could have been some of the applications, such as antenna theory, wavelets, time-frequency analysis, uses in Radar, speech processing... ... Read more

Reviews (1)

Willard's General Topology - a must for every bookshelf
One of the purest and most intellectually challenging branches of modern mathematics, general topology is not a subject for the faint hearted. So it was a pleasure when I first encountered one of the best reference introductions to the subject to have seen the light of day. Willard's book remains one of my all-time favourites. It covers everything the aspiring topologist needs to know, and certainly supplies more than enough information for a potential PhD student to choose their initial area of specialisation. The chapters are split intelligently into sub-topics which move at a sensible pace from its introductory notes on essential set theory, through subspaces, products, compactness, separation and countability axioms, compactifications, and function spaces. Many of the "standard spaces" of general topology are introduced and examined in the large number of related problems accompanying each section. And for those wanting a bit more context than a maths book normally provides there's a detailed collection of historical notes for each chapter. ... Read more

Book Description

This book describes many new results and extensions of the theory of generalized topological degree for densely defined A-proper operators and presents important applications, particularly to boundary value problems of nonlinear ordinary and partial differential equations that are intractable under any other existing theory. A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation.The theory subsumes classical theory involving compact vector fields as well as the more recent theories of condensing vector-fields, strongly monotone, and strongly accretive maps. Researchers and graduate students in mathematics, applied mathematics, and physics who make use of nonlinear analysis will find this an important resource for new techniques. ... Read more

Book Description

This volume covers the proceedings of an international conference held in Oxford in June 2002. In addition to articles arising from the conference, the book also contains the famous as yet unpublished article by Graeme Segal on the Definition of Conformal Field Theories. It is ideal as a view of the current state of the art and will appeal to established researchers as well as to novice graduate students. ... Read more

Book Description

Written by a computer scientist for computer scientists, this book teaches topology from a computational point of view, and shows how to solve real problems that have topological aspects involving computers.Such problems arise in many areas, such as computer graphics, robotics, structural biology, and chemistry.The author starts from the basics of topology, assuming no prior exposure to the subject, and moves rapidly up to recent advances in the area, including topological persistence and hierarchical Morse complexes. Algorithms and data structures are presented when appropriate. ... Read more

Book Description

This book combines mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation) in order to solve the conceptual and technical problems in the combining of elements of combinatorial and numerical algorithms.The book develops methods from areas that are amenable to combination and explains recent breakthrough solutions to meshing that fit into this category. It should be an ideal graduate text for courses on mesh generation.The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, are useful, and interesting. ... Read more

Book Description

In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe?It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one. ... Read more

Reviews (1)

Mainly for experts in Finsler geometry
The authors claim to turn the subject of Finsler geometry with this book
into a more teachable one and to have a candid style of writing.
This is definitly true for the first 50 pages, where the concepts of
Finsler geometry are very well explained and the exercises
are manageable and perfectly interrelated with the text.

Then the Chern connection and the curvature tensor of Finsler geometry
drop out of the heaven without any explanation of the ideas leading to these
constructions. So one has to derive them alone. In doing so older
texts on Finsler geometry like the Grundlehren text of Rund are more
helpful than this volume.

But the book was apparently prepared with great care. The layout must be called
beautyful and it really facilitates reading this book. Many references
to the literature and classical papers of the subject are included. So beginners,
which want to get a first aquaintance in Finsler geometry, find at least some help.

... Read more

Book Description

The Golden Section has played a part since antiquity in many parts of geometry, architecture, music, art and philosophy. However, it also appears in the newer domains of technology and fractals. In this way, the Golden Section is no isolated phenomenon but rather, in many cases. the first and also the simplest non-trivial example in the context of generalisations leading to further developments. It is the purpose of this book, on the one hand, to describe examples of the Golden Section, and on the other, to show some paths to further extensions. The treatment is informal and the text is enriched by the presence of very illuminating diagrams. Questions are posed at fairly frequent intervals and the answers to these questions, perhaps only in the form of very broad hints for their solution, are gathered together at the end of the text. ... Read more

Reviews (1)

A TOPIC TO TIE IN SEVERAL MATHEMATICAL SUBJECTS
The Golden Section is a very fascinating topic that is applied in several fields, both mathematical and nonmathematical. Why? Because this number primarily focuses on proportion, both in a physical and numerical context.

Somewhat to my chagrin, the author confines this subject within the realm of mathematics. For I wish that the Golden Section was discussed from a thematic or historical standpoint, such as how or why artists such as Da Vinci used this concept in order to construct or create works of art that were well-proportioned.

Nonetheless, the mathematics is delightfully presented. It is sketchy in some areas but is much easier to follow than several of the 50+-year-old-mathematical works that have not been revised or updated. Fascinatingly, this one topic alone synthesizes or ties in several mathematical disciplines, including: fractals, geometry, topology, number theory, trigonometry, and, strangely enough, probability.

I highly recommend this book. It is neat and concise, and it is a great reference for the mathematician who has been out of practice for a while to brush up on his or her skills. ... Read more

Book Description

This first edition of this book quickly became an established text in this fast-developing branch of mathematics. This second edition has been significantly revised and expanded. It includes a section on new developments and an expanded discussion of Taubes' and Donaldson's recent results. ... Read more

Reviews (1)

A must for researchers new to the field
An authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.

The discussion begins with classic topology and cover a variety of final year undergraduate topics such as complex manifolds and inverse differential techniques before moving into the vastly complex world of Symplectic Topology.

A must for researchers new to the field ... Read more

Book Description

This accessible introduction to harmonic map theory and its analytical aspects, covers recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A presentation of "exotic" functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a "Coulomb moving frame" is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces. ... Read more

Book Description

Over a century old, knot theory is today one of the most active areas of modern mathematics.The study of knots has led to important applications in DNA research and the synthesis of new molecules, and has had a significant impact on statistical mechanics and quantum field theory.

Colin Adams’s The Knot Book is the first book to make cutting-edge research in knot theory accessible to a non-specialist audience.Starting with the simplest knots, Adams guides readers through increasingly more intricate twists and turns of knot theory, exploring problems and theorems mathematicians can now solve, as well as those that remain open.He also explores how knot theory is providing important insights in biology, chemistry, physics, and other fields. The new paperback edition has been updated to include the latest research results, and includes hundreds of illustrations of knots, as well as worked examples, exercises and problems.

With a simple piece of string, an elementary mathematical background, and The Knot Book, anyone can start learning about some of the most advanced ideas in contemporary mathematics.
... Read more

Reviews (6)

Great introduction to knot theory
Having first been exposed to interesting knots while in undergraduate courses in biology and chemistry and occasionally encountering knots in my mathematical life, I have long maintained a passing interest in the field. However, until now, no single event evoked a reaction strong enough to pique a desire to explore. All it took to change that was the reading of this book by Adams.
Surprisingly complete for an introductory text, it is also amazingly understandable. Requiring only knowledge of polynomials and a mind capable of understanding twists, I found it addictive. This is one area where it pays not to think straight. After reading it twice, I still pick it up and scan it in odd moments. Problems are scattered throughout the book, and many can be solved using only a piece of string. Those that are still unsolved are clearly marked, with is good, since the statements are often very simple.
There are many applications and the number is growing all the time. One of the most profound images and statements of discovery was the pictures of the knotting of the rings of Saturn and commentator Carl Sagan saying, "We don't understand that at all. We will have to invent a whole new branch of physics to understand it." The most esoteric recent explanation of the structure of the universe is the theory of superstrings, where all objects are multi-dimensional knots. A fascinating problem in molecular biology of the gene is the process whereby DNA coils when quiescent and uncoils to be copied. One chapter is devoted to applications, although more would have been helpful.
A non-convoluted introduction to the theory of convolutions, this book belongs in every mathematical library.

Published in Journal of Recreational Mathematics, reprinted with permission.

Excellent motivation for knot theory
Knot theory has been a branch of mathematics that has been around for over a century, and now is finding applications in mnay areas, some of these being electrical circuit analysis, genetics, dynamical systems, and cryptography. This book, written for the layman or the beginning student of mathematics, is an excellent overview of what is known (and not known) in knot theory. Because of the pictorial nature of the subject, knot theory is an excellent way to get people interested in mathematics. Knot theory now is an established branch of mathematics, and it involves the use of tools from topology, analysis, and algebra. The problem of distinguishing one knot from another is one of the major questions in knot theory, and its partial resolution has been assisted by concepts from physics, namely statistical mechanics and quantum field theory. The author discusses the knot recognition problem, and other unsolved problems in the book, and he points out that in knot theory the unsolved problems can be approached by someone with very little background in advanced mathematical techniques. The author does an excellent job of introducing these problems and letting the reader experience, in his words, the joy of doing mathematics.

Chapter 1 is an introduction to the basic terminology of knot theory, and the author gives examples of the most popular elementary knots. He points out the historical origins of the theory, one of these being the attempt by Lord Kelvin to explain the origins of the elements, interestingly. The basic operations on knots are defined, such as composition and factoring, and the famous Reidemeister moves. The proof that planar isotopies and Reidemeister moves suffices to map one projection of a knot to another is omitted. After defining links and linking numbers, the author then discusses tricolorability, and uses this to prove that there are nontrivial knots.

Chapter 2 then overviews the strategies used in the tabulation of knots.The Dowker notation, used to describe a projection of a knot, is discussed as a tool for listing knots with 13 or less crossings. The author also discusses the Conway notation, and how it is used to study tangles and mutants. Graph theory is also introduced as a technique to study knot projections. The author discusses the unsolved problem of finding an elementary integer function that gives the prime knots with given crossing number, a problem that has important ramifications for cryptography (but the author does not discuss this application).

Since knots are complicated objects, then like many other areas in topology, the strategy is to assign a quantity to a knot that will distinguish it from all other knots. Such a quantity is called an invariant, and as one might guess, no one has yet found an invariant to distinguish all nontrivial knots from each other. In the last two decades though, new powerful knot invariants have been discovered, many of these being based on concepts from theoretical physics. In chapter 3, the author discusses the unknotting number, the bridge number, and the crossing number as elementary examples of knot invariants.

Chapter 4 is more complicated, in that the author shows how to use surfaces to assist in the understanding of knots. After discussing how to triangulate an surface and the concept of a homoeomorphism between surfaces, he introduces the Euler characteristic as an invariant of surfaces. Surfaces appear in knot theory as the space in the knot's complement, and the author introduces the concept of the compressibility of a surface, also very important in three-dimensional topology. Particular attention is paid to Seifert surfaces, which, given a particular knot, are orientable surfaces with one boundary component such that the boundary component is the knot in question.

Several different types of knots are considered in chapter 5, such as torus, satellite and hyperbolic knots. The latter are particularly interesting, since their study is part of the field of hyperbolic geometry, a subject that is now undergoing intense study. The author also introduces the theory of braids and the braid group. Not only are braids very important in the study of knots, but they have taken on major importance in cryptography and dynamical systems.

Chapter 6 is very interesting, and introduces some of the more contemporary topics in knot theory. The assignment of polynomials to knots goes back to the early 20th century, but it took the work of Vaughan Jones and his use of ideas from operator theory and statistical mechanics to provide polynomial invariants of knots that were much finer than the Alexander polynomial of the 1930s. The Jones polynomial however is not introduced the way Jones did, but instead via the Kaufmann bracket polynomial. The HOMFLY polynomial is introduced as a polynomial that generalizes the Jones and Alexander polynomials.

A few applications of knot theory are discussed in Chapter 7, such as the DNA molecule and topological stereoisomers. The author also discusses the applications of knot theory to the theory of exactly solvable models in statistical mechanics, a topic that has mushroomed in the past decade. This is followed by a brief overview of applications of knot theory to graph theory in chapter 8.

Chapters 9 and 10 are an introduction to knot theory as it relates to research in the topology of 3-dimensional manifolds and the existence of knots in dimensions higher than 3. The concepts introduced, particulary the idea of a Heegaard diagram, are used extensively in the study of 3-manifolds. In addition, the author mentions the famous Poincare conjecture, albeit in non-rigorous terms. The Kirby calculus, which is a kind of generalization of the Reidemeister moves, but instead models the sequence of operations that allow one to change from one Dehn surgery description of a 3-manifold to another is briefly discussed. The author also gives a few elementary, intuitive hints about how to visualize knotted objects in high dimensions.

Intelligent and intriguing!
I checked this book out of the library on the recommendation of a friend who was taking a knot theory class. While I am comfortable with calculus and differential equations, I have not had much experience with topology or group theory so I was hesitant. She assured me that I would understand the concepts presented there and that it would give a good introduction to the subject.

Wow! Was she ever right! First of all, the book is written in a clear and pleasant conversational style. The author does not hesitate to bring in examples or to show diagrams to clarify an idea. Indeed, with a subject such as knot theory, diagrams are essential! His use of exercises is well justified however, I would say that many laypersons are unfamiliar with proof techniques and thus might have some difficulties with several of those. Algebra is used sparingly at best as Adams prefers to let his words and images convey the ideas.

All in all, I would say that this book does a wonderful job of relating a subject which is at the forefront of mathematics, to the mathematically uninitiated. Hopefully, it will stimulate even further interest.

Owen

book is not that basic, provides direction
Book was purchased with the intent of getting direction into more complicated areas of knot theory, in particular adjacency matrices and using probability mean functions as a weighting technique of mapping.

searching the net for bookbinder's knot
trying VERY hard to find a bookbinder's knot "how-to" - help! please. where can i get a diagram so that i might learn? for the purpose of knitting a beaded purse, believe it or not! and need to use bookbinder's knot at edges. thank you. ... Read more

Book Description

Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds. A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics. ... Read more

Reviews (3)

Fun, but not very substantive.
This is a relatively fun romp through some very interesting concepts, but it lacks rigor.The book could have been much stronger if the author had simply developed some of the basic concepts (compactness, connectedness, homeomorphisms, homotopy, etc) rather than do a little hand-waving around a nice illustration.As it stands, this book is only 140 pages long, and does not develop any of its topics (manifolds, surfaces, graphs, knots) adequately.This book is far too weak to serve as a good text.Kinsey's TOPOLOGY OF SURFACES is much stronger, and costs less.Or look as something like Gamelin's INTRO TO TOPOLOGY.Or even Schaum's outline GENERAL TOPOLOGY, which deals with the basics, but is highly readable and rigorous.

Very Misleading Title, Quite Thin, No Rigor, and Overpriced
This book is subtitled "A First Undergraduate Course" but is certainly below undergraduate level.A high school student could easily follow this--which might be a good thing in certain cases--but the rigor is lacking.In fact, there is barely a hint of any rigor whatsoever.It is mostly intuitive arguments and the author often says things like "but we won't bother worrying about mathematical technicalities".It does get you to be able to visualize certain things well, but the visualization techniques can be found in other books also.The book is very thin and a quick read--hardly worth the money they are trying to get for it.If you're really at the undergraduate level and want to learn some topology, try something like Mendelson's "Introduction to Topology" by Dover or one of the excellent topology books in the series "Undergraduate Texts in Mathematics" by Springer.Munkres is also a classic.If you're not an undergraduate in a math related field and just want to know about the ideas behind topology or perhaps see some visualization techniques, try something like "The Shape of Space" by Weeks.Overall I was very disappointed with this text.If you could purchase this book for under $20 it might be worth it, but even then I think the other books I quoted are better in both price and substance.

No math library is complete without this book
This book presents the topology of surfaces, manifolds and knots in a manner that is reachable for undergraduate students with only a knowledge of calculus.Some linear algebra might be helpful.The text is written in a style that is easy to follow and there are superfluous examples.The exercises in the text are well thought out and are not extremely difficult.The exercises complement the text very well.The text makes clear a lot of difficult concepts such as isotopic surfaces as opposed to homeomorphic surfaces.I particularly enjoyed the manner in which the topology of knots was explained.After reading this text, the reader should be able to better visualize the projective plain and even the Klein bottle as it exists in 4-dimensional space.I have not read a text on topology that I enjoyed reading as much since Munkres.This text is a must have for any topologist. ... Read more

Book Description

This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow.

The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgangers-that is, other objects that are extremely difficult to distinguish from it.

Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems," and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.

... Read more

Book Description

A new approach to understanding nonlinear dynamics and strange attractors
The behavior of a physical system may appear irregular or chaotic even when it is completely deterministic and predictable for short periods of time into the future. How does one model the dynamics of a system operating in a chaotic regime? Older tools such as estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions do not sufficiently answer this question. In a significant evolution of the field of Nonlinear Dynamics, The Topology of Chaos responds to the fundamental challenge of chaotic systems by introducing a new analysis method-Topological Analysis-which can be used to extract, from chaotic data, the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic data. Beginning with an example of a laser that has been operated under conditions in which it behaved chaotically, the authors convey the methodology of Topological Analysis through detailed chapters on:
* Discrete Dynamical Systems: Maps
* Continuous Dynamical Systems: Flows
* Topological Invariants
* Branched Manifolds
* The Topological Analysis Program
* Fold Mechanisms
* Tearing Mechanisms
* Unfoldings
* Symmetry
* Flows in Higher Dimensions
* A Program for Dynamical Systems Theory
Suitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this groundbreaking approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems. ... Read more