Product Description
Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.

Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. ... Read more

Customer Reviews (13)

Excellent review of math for (particle) physicists
I bought this book to supplement my knowledge of mathematics which frequently is involved in understanding Particle physics concepts. The book is terse, but peppered with examples and insights about the definitions, and so far it is really fun to read. Seems like a good investment.

Too many errors to be useful for study
Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page.
The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.

Geometry Topology and Physics: A condesed view
This book provide a complete and useful review of geometrical instuments of mathematical physics from the beginnig to the most advanced topics of interest. It can be used by students at the beginnig of thei studies in this topics, and it's found to be a useful gallery for higher level students (or scholar).

An excellent book
This is the best book of its type, that is, a book that contains almost all if not all the advance mathematics a theoretical physicist should know. I have studied chapters 2-9 and it has the perfect balance between rigorous presentation of topics and practical uses with examples. The level is for advance graduate students. The range of topics covered is wide including Topology topics like Homotopy, Homology, Cohomology theory and others like Manifolds, Riemannian Geometry, Complex Manifolds, Fibre Bundles and Characteristics Classes. I believe this book gives you a solid base in the modern mathematics that are being used among the physicists and mathematicians that you certainly may need to know and from where you will be in a position to further extent (if you wish) into more technical advanced mathematical books on specific topics, also it is self contained but the only shortcoming is that it brings not many exercises but still my advice, get it is a superb book!

A great reference book.
No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics. ... Read more

Product Description
Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. ... Read more

Customer Reviews (6)

An Introduction with Mathematical Integrity
Gregory Naber is to be commended for writing a thorough introduction to gauge field theory in which the mathematics is presented with clarity and rigor.For the professional mathematician who is interested in physics, or for the graduate student who prefers to see the mathematics "done right" in advanced applications to physics, Naber's wonderful two-volume set stands apart from its major competitors, nearly all of which were written by physicists, for physicists.

Despite the attention to mathematical rigor, it is clear that Naber intended his books to be accessible to a dual audience of physicists and mathematicians. For the physicists, he has included gentle introductory chapters on topological spaces, homotopy groups, principal bundles, manifolds and Lie groups, and differential forms. For mathematicians, the chapters on physical motivation, gauge fields and instantons, Yang-Mills-Higgs theory, Spinor structures, etc., provide unusually accessible introductions to some difficult physics materials.

Chapter 0 of the first volume is worth the price of both books, as it leads the reader, in 26 succinct pages, to a compelling appreciation ofthe natural "fit" of the Hopf Bundle to the task of providing a quantum mechanical analysis of the exterior of a single magnetic monopole.For outsiders who have become incredulous about the increasingly sophisticated uses of topology and geometry in theoretical physics, this example provides some much-needed assurance.As the reader quickly learns, the use of connections on principal fiber bundles is neither gratuitous nor mathematical overkill:indeed, the bundle machinery emerges quite NATURALLY as the simplest and best mathematical tool, perfectly fitted to the special problem at hand.

Any serious reader will want to buy both volumes of this set:Topology, Geometry, and Gauge Fields:Foundations (volume 1), and Topology, Geometry, and Gauge Fields:Interactions (volume 2).These books take their place alongside the work of authors such as Jerrold Marsden, Theodore Frankel, Barrett O'Neill, and Walter Thirring, all of whom write about modern mathematical physics in a way that does not obscure the true role of the mathematics.

correction to dost
The review "Easy reading, complete proofs, plenty of exercises, October 29, 2005 by Rehan Dost is of the first volume, Foundations, not this volume which is Interactions. Naber's books are crafted to bridge physics, undergraduate mathematics and graduate mathematics. This is one more of his beautiful volumes in applied mathematics.

Easy reading, complete proofs, plenty of exercises
This text is by far the best introductory text marrying basic concepts of physics with pure mathematics.

Some background in the basic concepts of vector calculus, linear algebra, complex numbers and group theory is required.

The author begins by motivating the mathematics by the pursuit of finding a vector potential to represent a magnetic monopole. We see that the topology of R3-0 precludes such a vector potential from existing. We see here a simple example of how the topology of a space affects the physics associated with it.
The importance of the vector potential as something other than a convenient computational tool is highlighted by a reference to essential inclusion in quantum mechanics. Thus we NEED such a potential.

The author now asks whether there is a "trick" or device to get around this difficulty. The device are principal bundles and connections. For example the potentials noted above must keep track of the phase of a charged test particle as it moves thru the field of a magnetic monopole. We need a "bundle" of circles ( representing the phase at each point ) over S2 ( the author explains why we need only consider S2 instead of R3-0, briefly we need only keep track of 2 of the 3 spherical co-ordinates ).
Thus a curve in S2 thought of as the particles trajectory will have to be "lifted" to the bundle space by a lifting procedure called a connection.
In a more general setting elementary particles have an internal structure ( spin etc ) which becomes apparent during interactions although may not be apparent in uniform motion thru a vacuum. Since the phase of the particle does not alter the modulus when calculating probabilities these do not change. However, when the particles interact phase differences are important. We need to keep track of such phases as the particles interact.

Thus we need a "bundle" over a 4-manifold ( keeps track of the particles space-time path ) to keep track of such internal states. One sees we also need a group to transform states into one another ( usually incorporated into the bundle ). Connections then model physical phenomena which mediate changes in the internal states.
We see that some connections satisfy the Yang-Mills equations and using the appropriate equivalence relation form Moduli spaces.

Now that may seem like alot to digest with only a spattering of mathematical maturity.

The beauty of the book is that the author starts from FIRST principles.

Chapter 1 introduces topological concepts of topology, continuity, quotient topology, projective spaces, compactness, connectivity, covering spaces and topological groups.

Chapter 2 introduces concepts of path lifting, fundamental groups, contractability, simple connectedness, covering homotopy theorem, higher homotopy groups

Chapter 3 introduces principle bundles, transition functions, bundle maps and principle bundles over spheres.

Chapter 4 introduces manifolds, derivatives on manifolds, tangent/cotangent spaces, submanifolds, vector fields, matrix lie groups, vector valued 1- forms, 2 forms and Riemann metrics

Chapter 5 gets to some physics with gauge fields and connections, curvature, Yang-Mills functional, moduli spaces, Hodge dual , matter fields and covariant derivatives.

At each step the author carefully provides complete proofs and easy exercises to ensure understanding.

It was a pleasure to read the book and complete the exercises. At no point did I feel frustration or boredom.

Don't waste your money
This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

MATH AND TOPOLOGY
Topology is very important scince in the fields of mathematics. And it using in many of another sinceis. ... Read more

Product Description

This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory.

From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS

Customer Reviews (7)

never really liked it
but maybe you will.
i think it was Vick's book that i preferred.
Massey--I preferred Massey.
i think that this is the one for the grown-ups.

a different perspective
While I agree with reviewers generally that this is a good book, i should warn that bredon isnt for the faint of heart. He makes use of simple language from category theory, doesnt always completely introduce his discussions (see for example the chapter on the tangent bundle where tangent bundle is never defined), and some other things that are nuisances to the newcomer.

I do think this is a good modern readable textbook, but for the student who has a solid foundation in mathematics. I didnt find it as accessable as other topology books, say Hatcher or Lee's books (but lee's are not as complete).

excellent for first year graduate study
This was the assigned book in my first year grad topology course. It has good examples, interesting exercises. I like the emphasis on geometrical examples, constructions. It's not easy to read, but interesting.

Among the best textbooks in algebraic topology.
As the previous reviewers have commented, this book is very accessible and detailed. I should add that the authour never lets you get lost in the labyrinth of abstract nonsense; the treatment is always geometric rather than homologico-algebraic. The only complaint I have is, the book would be more useful with chapters on spectral sequences, cobordism and K-theory.

The Graduate Sudent's Topology Bible
If you want to learn topology, this book is the place.Though this text can require some maturity, the range of topics and the clarity of exposition are outstanding.My only complaint is that an additional appendix covering the basics of category theory would have been quite useful.Bredon not infrequently uses the language of category theory (though always in a non-essential way).Since this text is aimed at 1st year graduate students, I think the tacit assumption that the student has already encountered these topics is not justified.That such a minor point is my chief complaint speaks volumes of my esteem for this text. ... Read more

Product Description
This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.
... Read more

Customer Reviews (3)

Perfect book for the right reader
This book is not a textbook -- nor is it a rigorous buildup of topology from geometry. It is simple and conceptual bridge from the concepts of congruence classes in geometry, and basic ideas from map theory, into topology.
For any person wishing to acquire a grasp of the subject, I would recommend this book as a primer to any more conventional treatment of topology. In many texts, you are given general theorems in terms of sets, with no real idea of why these ideas have come about. From Geometry to Topology provides the necessary intuitive background... which is not to mention -- it's a darn interesting read.

Easy, but okay...
This book is quite pictorial and thus easy to read, as it's intended to be: a first introduction on undergraduate level to topology. However, it's a pity it doesn't go into topology very much. It stays very informal.

Very nice and intuitive introducation to topology
I bought this book shortly after my introductory analysis professor had once mentioned what is topology. That was the first topology book I had read, and I fell in love with the subject ever since. It's easy to understand,and very enlightening as well.
The book walks you through the transition from geometry to topology, then eleborates on several basic topological concepts.
Very interesting stuff ! ... Read more

Product 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

Customer Reviews (6)

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.

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 conceptsof 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. ... Read more

Product Description
The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R3 . The material here is accessible to math majors at the junior/senior level. ... Read more

Customer Reviews (5)

Great Book
It is a very intiutive book in both areas. Also at the end of the book there is a good material for further study, author explains the research fields in Geometry/Topology and related books. If you are an undergraduate and want to get an overall idea about the gradute study in topology and geometry that is a nice introduction.

a remark on omissions
I have not read the book, only the reviews.In one excellent review here it is remarked that it is "unfortunate" that the author does not prove the Schoenflies theorem and the triangulability of surfaces.

later this same reviewer observes that the proof of the smooth Gauss Bonnet theorem in the book seems relatively hard.I merely wish to point out that the author has made choices in the reader's interest both by what he includes and what he omits.

The two theorems named above which are not proved, could well take another entire book to prove.They are far harder than the smooth Gauss Bonnet theorem.

I have seen entire books devoted to proving triangulability, and Schoenflies theorem was the subject of weeks of tedious work in a topology course I took as a student.I still dislike even hearing of this result.So if these omissions are the reviewer's only criticisms of the book, they should rightly be considered pluses.

Hence I also give the book at least 4 stars, by logical deduction.

Good introduction
This book is suitable for reading at an advanced undergraduate or beginning graduate level. The author is careful to present the subject from both a rigorous point of view and one that emphasizes the geometric intuition behind the subject. These two approaches to teaching topology are not mutually exclusive, with this book giving a good example of this.

After a brief overview of the elementary topology of subsets of Euclidean space in chapter 1, topological surfaces are discussed in chapter 2. Surfaces are built up from arcs, disks, and one-spheres. Unfortunately, the proofs of the theorem of invariance of domain and the Schonflies Theorem are not included, but references are given. Gluing techniques though are effectively discussed, and the author does not hesitate to use diagrams to explain the relevant concepts. The more popular constructions in surface topology, namely the Mobius strip and the Klein bottle are given as examples of the cutting and pasting techniques. The amusing fact that the Klein bottle can be obtained from gluing two Mobius strips along their boundaries is proven.

The theory of simplicial surfaces is discussed in the next chapter. Simplicial surfaces are much easier to deal with for beginning students of topology. Simplicial complexes are introduced first, and the author then studies which simplicial complexes have underlying spaces that are topological surfaces. He proves that this is the case when each one-dimensional simplex of the complex is the face of precisely two two-dimensional simplices, and the underlying space of each link of each zero-dimensional simplex of the complex is a one-dimensional sphere. Unfortunately, the author does not prove that any compact topological surface in n-dimensional Euclidean space can be triangulated. The Euler characteristic is defined first for 2-complexes and it is shown that it is the same for two simplicial surfaces that triangulate a compact topological surface. The author does prove in detail the classification of compact connected surfaces. Interestingly, the author also proves a simplicial analogue of the Gauss-Bonnet theorem, and gives a proof of the Brouwer fixed point theorem.

The author turns to smooth surfaces in the next few chapters, wherein curves are defined along with the relevant differential-geometric notions such as curvature and torsion. The fundamental theorem of curves is proven. The reader is first introduced to the concept of what in more advanced treatments is called a differentiable manifold, and several concrete examples are given of smooth surfaces. The differential geometry of smooth surfaces is outlined, with the first fundamental form and directional derivatives discussed in great detail. The reader should be familiar with the inverse function theorem to appreciate the discussion of regular values.

Even more interesting differential geometry is discussed in chapter 6, which covers the curvature of smooth surfaces. The important Gauss map is defined, along with the Weingarten map and the second fundamental form. This allows an intrinsic notion of curvature, but the author does perform explicit computations of curvature using various choices of coordinates. The proof that Gaussian curvature is intrinsic (Theorema Egregium) is proven, along with the fundamental theorem of surfaces. Geodesics, so important in physical applications, are discussed in the next chapter. The reader gets a first look at the "Christoffel symbols", even though they are not designated as such in the book.

The book ends with a thorough treatment of the Gauss-Bonnet theorem for smooth surfaces. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. The author also gives a very brief introduction to non-Euclidean geometry.

Presentation of The Spirit...
Lots of times the mathematicians stuck in proofs,in that fool symbols, forgetting the ideas, the picture.One can never find the right way withclosed eyes.This book teaches to think, getting beyond the symbols. Ithas also useful advises about the research areas. The author made his Phdat Cornell with D.Henderson.A beautiful undergraduate text.

Just a mediocre book of lesser extent
First, the title reads fine. But there's a catch. This kind of title sounds like it covers all. The truth is. It ain't true. Second. The author's attitude. I'd rather say the author is talking to himself. ... Read more

Product Description
This book develops some of the extraordinary richness, beauty, and power 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 living feeling for the subject. There are many figures, examples, and exercises of varying difficulty. ... Read more

Customer Reviews (4)

a decent book, but
FAR worse than Thurston's classic notes, whence it came. Covers very little material by comparison, and has a lot more bogus "explanations". Given that the notes are available for free from Math. Sciences Research Inst. (MSRI), in convenient PDF form (for us Kindle DX Wireless Reading Device (9.7" Display, Global Wireless, Latest Generation)and Apple iPad MB292LL/A Tablet (16GB, Wifi) users, this is not a great book to get.

This book is the real deal by comparison
I have this book by Jeffery Weeks that Thurston's opens up with the real mathematics: The Shape of Space (Pure and Applied Mathematics). In both cases though, they fail to make the connection of the higher Cartan Lie Algebras to 3 manifold theory clear. When the connection of string theory to 3 dimensional geometry comes through this
approach to geometry, you think that we have a theory that neglects 5 dimensions and above?
So as good as this text is, it still falls somewhat short of what is needed in the modern world and we are forced to think of our own way to interpret the standard model symmetry breaking of SU(5) (CartanA_4) to U(1)*SU(2)*SU(3).

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 isreadable to undergraduates.Its target audience, though, is beginninggraduate students in mathematics.If not already familiar with hyperbolicgeometry, you might want to get an introduction to the subject first.Oncewith this background, though, you will discover there is another level ofunderstanding of hyperbolic space you never realized was possible.Oneimagines Thurston able to skateboard around hyperbolic space with the kindof geometric understanding he conveys here.

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

Product Description

This book covers a new explanation of the origin of Hamiltonian chaos and its quantitative characterization. The author focuses on two main areas: Riemannian formulation of Hamiltonian dynamics, providing an original viewpoint about the relationship between geodesic instability and curvature properties of the mechanical manifolds; and a topological theory of thermodynamic phase transitions, relating topology changes of microscopic configuration space with the generation of singularities of thermodynamic observables. The book contains numerous illustrations throughout and it will interest both mathematicians and physicists.

Product Description

From reviews of the first edition:

"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."

Mathematics Teaching

"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."

Christoph Bandt, Mathematical Reviews

"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

Customer Reviews (4)

some new material->same approach
I bought the first edition of this in the early 90's
and was disappointed that it didn't have the Mandelbrot or other complex dynamics
in it. Dr. Edgar has updated the older book with Julias, multifractals
and Superfractals, but has stayed true to his topological measure theory
Hausdorff space approach. He never updates his Biscovitch-Ursell functions
to 2d and 3d parametrics or the unit Mandelbrot cartoon method.
Some of his definitions are still so minimal
that duplicating the fractals needs much more information?!
The text is still the good place to begin, but
it is a shame that Dr. Edgar has not kept up
with many of the developments in the field. Zipf and Per Bak
are left out, but my double V L-system made the index as a picture.

A nice book
I liked this book because it provided me with a new perspective on metricspaces, in using them as a basis learning about fractals. I think it servesas a nice book for an undergraduate to read and get enthused about studyingfractals at a higher level.

Good starting point to study fractal geometry.
This book could be used as a bridge between traditional books on topology-analysis and the speciallized treatises on fractal geometry. More a catalog of definitions, methods, and references than a course text, itcovers the fundamental topological and measure-theoretic concepts needed tounderstand the principles of some of the different dimension theories thatexist. But warning: the book is far away of being a complete exposition onany of the subjects it includes.

Suitable for 3rd-year undergrads.Interesting examples and exercises. Extensive bibliography.

Please checkmy other reviews in my member page (just click on my name above).

A difficult but worthy book!
The programs are in LOGO: don't let the turtles fool you, this is the real stuff by a master teacher. It is hard and the examples are even harder. The problem sets are at times impossible, but in the end Dr. Edgar delivers: understanding! Your unique Associates ID is:thefractaltransl. ... Read more

Product Description
Geometry and topology are strongly motivated by thevisualization of idealobjects that have certain specialcharacteristics. A clear formulation of a specific propertyor a logically consistent proof of a theorem oftencomesonly after the mathematician has correctly "seen" what isgoing on. These pictures which are meant to serve assignposts leading to mathematical understanding, frequentlyalso contain a beauty of their own.The principal aim of this book is to narrate, in anaccessible and fairly visual language, about some classicaland modern achievements of geometry and topology in bothintrinsic mathematical problems and applications tomathematical physics. The book starts from classicalnotionsof topology and ends with remarkable new results inHamiltonian geometry. Fomenko lays special emphasis uponvisual explanations of the problems and results anddownplays the abstract logical aspects ofcalculations. Asan example, readers can very quickly penetrate into thenewtheory of topological descriptions of integrableHamiltoniandifferential equations. The book includes numerousgraphicalsheets drawn by the author, which are presented inspecialsections of "Visual material". These pictures illustratethemathematical ideas and results contained in the book. Usingthese pictures, the reader can understand many modernmathematical ideas andmethods.Although "Visual Geometry and Topology" is aboutmathematics,Fomenko has written and illustrated this bookso that students andresearchers from all the naturalsciences and also artists and art students will findsomething of interest within its pages. ... Read more

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This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.

Customer Reviews (4)

Best Intro to Topology
Topics are well motivated.
Theorems are proved in a rigorous yet intuitive style that one feels like it was an explanation rather than a dry proof typically found in the advanced math books.
Important key ideas are also sufficiently illustrated through examples and exercises.

If one finds it verbose, I'd recommendcroom--a bit more like the typical math books but accessible.

Surely not optimal
We used this book in an introductory topology class I took.Some of the exercises are poor (e.g. counting the number of topologies) and the exposition wasn't anything to go crazy about.After a while, I found myself reading Munkres exclusively; it's much more comprehensive.Maybe this book is well suited for folks looking to get a flavor of topology but nothing super concrete.

Best undergraduate topology book
I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

A pleasure to read
I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide. ... Read more

Product Description
Contents: Introduction. - Fundamental Concepts. -Topological Vector Spaces.- The Quotient Topology. -Completion of Metric Spaces. - Homotopy. - The TwoCountability Axioms. - CW-Complexes. - Construction ofContinuous Functions on Topological Spaces. - CoveringSpaces. - The Theorem ofTychonoff. - Set Theory (by T.Br|cker). - References. - Table of Symbols. -Index. ... Read more

Customer Reviews (5)

great as motivation but not a textbook
While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathematicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the author even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.

Certainly this book couldn't be used as a textbook for an undergraduate course - for the reasons mentioned above and also because not enough material is actually covered, as well as the obvious deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated. The book also covers both point-set topology (topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.) and elements of algebraic topology (homotopy, fundamental group, simplices, CW-complexes, covering spaces, but not really homology), but the presentation of the algebraic topology in particular is not liable to be helpful for the novice, except for the treatment of covering spaces, which is perhaps the highlight of the book. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index:
"We talk about homology (and a number of other objects beyond the realm of point-set topology) several times in this book, but the definition is not given."
That, in a nutshell, explains the difficulty with this book.

So why am I rating this 5 stars? For the wealth of examples (e.g., 4 sections on examples of quotient spaces) and explanations of how these concepts are used and why they are important. Just by looking at the contents one can see this, as there are sections titled:
"What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?," "The role of the countability axioms," "Why CW-complexes are more flexible," "Yes, but...?," "The role of covering spaces in mathematics," "What is it [Tychonoff's theorem] good for?"
The chapter on covering spaces, coming near the end of the book, is particularly good, with a proof of their classification given. This is definitely the most fleshed-out part; if only the rest of the book could go into this depth.

This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a substitute for it. But I would still consider this as a must-read for all those students who plan on studying mathematics in graduate school.

A simple introduction to advanced mathematical concepts
This text gives the reason behind many advanced topological concepts and tantalizes the reader with it's varied applications.

Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.

The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).

Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.

I especially enjoyed the section on covering spaces with which it concludes.

Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.

The text is useful to all students of mathematics and physics alike.

Full of motivations
This book is fun to read. In a weekly homework meeting for an Algebraic Geometry class, I complained to one grad student "Geometry textbooks should have many pictures", and he asked "Define 'many'?" I said "One on each page". Now this topology book is certainly close to that. (It has more than 180 illustrations.) Though its written style is a bit informal, 'handwaving' arguments can serve as outlines of rigorous proofs.

Since it does not have any problem sections, I can see why Munkres' book is more popular in college. It still gives some inspiring questions from time to time. Besides the basic pot-set topology, it also covers some algebraic topology and differentialtopology. The author does not hesitate to use examples from those advanced areas without formal definitions, and this was a bit annoying when I read it the first time. In this sense, the book is not really selfcontained. However, when finally a notion is formally defined, I can see it from various aspects in those examples. This really helps me understand topology better, and makes me want to explore them. After reading the existence thm of covering spaces in chapter 9, I realized that mathematics is really an art.

The index in the back of the book is in the format of short definitions, which can be used as a quick reference.

Students: BUY THIS BOOK!!!
It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to "see into the future" of what kind of mathematics is waiting for them and how topology is indispensable in its study.

At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: "If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just 'handwaving'. Shall we then abandon all intuitive arguments? Certainly not. As long as it is backed by the gold standard of rigorous proofs, the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Long live handwaving!". This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, "playful" understanding of the mathematical constructions.

Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.

If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK...BUY AND STUDY THIS BOOK. You will take away so much for the price paid.

Excellent
Excelent, clear, well-motivated introduction ... Read more

Product Description
Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.

Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models.

The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.

The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field. ... Read more

Customer Reviews (2)

Excellent book
It was a great pleasure to read the book “Differential Geometry and Topology With a View to Dynamical Systems” by Keith Burns and Marian Gidea. The topic of manifolds and its development, typically considered as “very abstract and difficult”, becomes for the reader of this outstanding book tangible and familiar.This joyful aspect of the book was achieved by the authors by setting the advanced material of differential geometry and topology as if on a “mobile bridge” or a “crossroad” that associates a(n) (primarily) unfamiliar abstract part of the text with elementary math theories. The latter pedagogical approach was mostly carried out through carefully prepared examples, in which, for essentially abstract structures and mathematical topics, well known familiar elementary settings serve as obvious motivations, which make the transition to a higher level of an abstraction smooth. Nevertheless, the scope of the main topic in this book, differential geometry and topology, is pretty far advanced.Besides the basic theory, centered around analytical properties of manifolds (mostly endowed with additional, in particular Riemannian, structures and vector or tensor fields defined on them) and their applications, it also provides a good introductory approach to some deeper topics of differential topology such as Fixed Points theory, Morse theory, and hyperbolic systems throughout the rest of the book.
The main stream of the applications that always follow or motivate the theoretical context is dynamical systems. Excellent examples reveal the close ties ofthisbeautiful mathematical theory with common problems intheoreticalphysics, classical and fluid mechanics, field theory, and, most importantly, the theory of general relativity.
The book by Burns and Gidea is also be strongly recommended for those readers who wish to enhance their mathematical tools to make possible a deeper insight into these fascinating physical theories.

Jerzy K.Filus

A very good book
A very clear and very entertaining book for a course on differential geometry and topology (with a view to dynamical systems).

First let me remark that talking about content, the book is very good. Each of the 9 chapters of the book offers intuitive insight while developing the main text and it does so without lacking in rigor. The first 6 chapters (which deal with manifolds, vector fields and dynamical systems, Riemannian metrics, Riemannian connections and geodesics, curvature and tensors and differential forms) make up an introduction to dynamical systems and Morse theory (the subject of chapter 8). Chapter 7 is devoted to fixed points and intersection numbers. The last chapter is an introduction to hyperbolic systems.

This enjoyable and highly instructive book contains a large number of examples and exercises. It is an incredible help to those trying to learn dynamical systems (and not only). It teaches all the differential geometry and topology notions that somebody needs in the study of dynamical systems.
The authors, without making use of a pedantic formalism, emphasize the connection of important ideas via examples. It completely enhanced my knowledge on the subject and took me to a higher level of understanding.

... Read more

Product Description
This volume presents an array of topics that introduce the reader to key ideas in active areas in geometry and topology. The material is presented in a way that both graduate students and researchers should find accessible and enticing. The topics covered range from Morse theory and complex geometry theory to geometric group theory, and are accompanied by exercises that are designed to deepen the reader's understanding and to guide them in exciting directions for future investigation. ... Read more

Product 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

Customer Reviews (2)

A very nice, clear book
And much more pleasant to read than Edelsbrunner's Computational Geometry book. I have very little interest in Mesh Generation, but most of the book is not about that, but is rather a rather lucid introduction to central topics in modern discrete and computational geometry.

Excellent read for mathematically inclined reader
OK, I must admit, I know the author personally, like his work a lot. And so I can only recommend this book. But that's not what you're looking for in a review, nor am I talking to freinds or collegues who already know hime too.This book will introduce you to simplicial complexes, and deep mathematical constructs, along with some topology and geometry, while at the same time remaining hands-on and simpler.Mostly, the notation is clean, simple, and yet rigorous enough that you'll really be in terrific shape one you integrate it. It's actually amazing that it comes so clean given how powerful it can be. Meshes are the basis for many of the computer graphics and CAD/CAM modern methods, and are an indispensible tool. That's the real value of reading this book: you'll get some real good tools for manipulating meshes (whether what you want to do is the same or different than the author), and especially you'll get a mathematically correct and rigorous treatment.

At the same time this book is quite manageable even (and foremost) if you don't have a PhD in algebraic topology (all you need is a good bachelor in some computational science with good mathematics foundations). Although it might challenge you at times, it is basically self-contained and does not rely on any other book. (Additional knowledge is always useful, but this is a good starting point.)
You'll earn about simplicial complexes not from an abstract algebraic topology point of view (although the author is well-acquainted with them) but as a tool for representing surfaces and solids.The topics revolve around reconstructing a surface from a point cloud (using so-called Delaunay triangulations, which is one of the prevalent methods for that problem). It starts simply with 2D and moves on to 3D.It builds on the research of the author for more than a decade.

I should mention in fine that the author is the creator of Alpha-Shapes(a well known Delaunay-based method for reconstruction) and of the succesful startup Raindrop Geomagic which uses these methods in an essential way. His current interests are linked to biogeometry. Most of the research in this book somehow made it in one way or the other in their products. So this is excellent reading and a must for anyone interested in meshes, whether from computer graphics, CAD/CAM, or scientific computation / finite elements. ... Read more

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Application of the concepts and methods of topology and geometry have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity and particle physics. This book can be considered an advanced textbook on modern applications and recent developments in these fields of physical research. Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, sypersymmetric solitons and noncommutative geometry. It will be of benefit to postgraduate students, educating newcomers to the field and lecturers looking for advanced material.

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Geometry is central to many branches of mathematics and physics, and offers a complete range of views on the universe. This introduction includes many simple explanations and examples. With minimal prerequisites, the book provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. The book is based on many years' teaching experience, and is thoroughly class-tested. There are copious illustrations, and each chapter ends with exercises. Further teaching material is available via the web, including assignable problem sheets with solutions. ... Read more

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This volume discusses results about quadratic forms that give rise to interconnections among number theory, algebra, algebraic geometry, and topology. The author deals with various topics including Hilbert's 17th problem, the Tsen-Lang theory of quasi-algebraically closed fields, the level of topological spaces, and systems of quadratic forms over arbitrary fields. Whenever possible, proofs are short and elegant, and the author has made this book as self-contained as possible. This book brings together thirty years' worth of results certain to interest anyone whose research touches on quadratic forms. ... Read more

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In the Spring of 1984, the authors gave a series of lectures in the Institute for Advanced Studies in Princeton. These lectures, which continued throughout the 1984-1985 academic year, are published in this volume. Lectures on Differential Geometry was originally printed in Chinese, and widely circulated in China. This greatly anticipated English translation is an essential reference tool for Differential Geometry. Differential Geometry has progressed rapidly in this century. This book describes the major achievements in this field. ... Read more