The Shape of Space (Pure and Applied Mathematics)

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Maintaining the standard of excellence set by the previous edition, this textbook covers the basic geometry of two- and three-dimensional spaces Written by a master expositor, leading researcher in the field, and MacArthur Fellow, it includes experiments to determine the true shape of the universe and contains illustrated examples and engaging exercises that teach mind-expanding ideas in an intuitive and informal way. Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhäuser Classics)

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profoundly important mathematics
The nature of high dimensions: a geometric insight

Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhäuser Classics)
Mikhail Gromov , M. Katz , P. Pansu , and S. Semmes
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Paperback

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ASIN: 0817645829

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Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory.

The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov.

The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov–Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy–Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity.

The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous "Green Book" by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices—by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures—as well as an extensive bibliography and index round out this unique and beautiful book.

Customer Reviews:

profoundly important mathematics.......2006-10-27

This book (originally published in French and improved here) is a fundamentally important book opening up an entire field of mathematics. For a textbook based on this material and related topics try Burago-Burago-Ivanov's textbook on the subject which can be taught to first year graduate students. For mathematicians and advanced graduate students, Gromov's book is a masterpiece.

The nature of high dimensions: a geometric insight.......2000-02-27

Formally speaking, this is the second edition of a set of Paris lecture notes published by Gromov two decades ago in the French language. However, such a wealth of entirely new material has been added that in essence we are talking of a new book.

Among the additions, the bulky new chapter 3 1/2+ stands out, dealing with the phenomenon of concentration of measure on high-dimensional structures. This is a relatively recent discovery of modern analysis and geometry, tracing its origin to the work of Paul Levy and especially Vitali Milman. The essence of the phenomenon is that on many multidimensional structures, every `nice' function is constant with high probability. The manifestations of the phenomenon are many - from geometric functional analysis (Dvoretzky theorem) through information theory (blowing-up lemma) and probability (law of large numbers) to graph theory (superconcentrators) and topological dynamics. As Gromov stresses in his book, even deeper aspects of the concentration phenomenon have been long since discovered and are constantly explored in statistical physics in the context of phase transitions of various kind, and some of the first known examples where phase transitions appear in the context of geometry have been discovered by Gromov himself, e.g. for hyperbolic groups. Finding and exploring more instances of phase transitions in mathematics might well become a unifying heuristic principle across a large number of disciplines.

The mathematical setting for dealing with concentration and related issues is the concept of a metric space equipped with finite measure, what Gromov calls an mm-space. Apart from concrete objects (such as for instance spheres and cubes), there are `higher-level' examples of mm-spaces, for instance those whose elements are isomorphism classes of mathematical objects themselves (e.g. Riemanning manifolds or finitely generated groups). This leads to a probabilistic treatment of such objects. Of course Gromov's strength is that his treatment is always concrete and he never theorizes without having particular objects and applications in mind.

It is quite safe to claim that the full range and power of applications of the interaction between metric and measure are yet to be discovered, which is what makes this book so important. It is rich in open questions and suggested new research directions, but more than that, it helps the reader to develop a good intuitive feeling of where things are going these days, what things ought to be done, and what constitutes proper mathematics.

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 doppelgängers--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.

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Slow but good
Technically great, but ultimately cold and depressing
MANIFOLD ZZZZZZZZzzzzzzz
Hope for our future
Manifold arghhhhhh!

Manifold: Space (Manifold)
Stephen Baxter
Manufacturer: Del Rey
ProductGroup: Book
Binding: Mass Market Paperback

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ASIN: 0345430786
Release Date: 2002-01-02

Amazon.com

Stephen Baxter follows up his Arthur C. Clarke Award nominee Manifold: Time with the second book in the Manifold series, Manifold: Space. In this novel, former shuttle pilot and astronaut Reid Malenfant meets his destiny once again in a tale that stretches the bounds of both space and time.

The year is 2020 and the Japanese have colonized the moon. The 60-year-old Malenfant is called there by a young scientist named Nemoto who has discovered something in the asteroid belt that can only mean humans are not alone in the universe. The aliens seem robotic in nature and appear to be building something in Earth's backyard. The Gaijin, as they are called by humans, don't respond to communication efforts so an unmanned ship is launched to investigate. In the meantime, Malenfant decides answers are only possible by mounting an expedition to Alpha Centauri, which may be where the Gaijin come from.

Baxter, who won the John W. Campbell Award and the Philip K. Dick Award for his novel The Time Ships, orchestrates a stunning array of scientific possibilities in Manifold: Space. Each chapter adds a new piece to his mosaic of humanity's future. The novel is admirable in its enormous scope, but it's hard to invest much emotion in the characters. Although they are well drawn, they vanish for long periods of time as Baxter leapfrogs through time and space. Manifold: Space, by its nature, lacks passion but excels in grand ideas. --Kathie Huddleston

Book Description

The year is 2020. Fueled by an insatiable curiosity, Reid Malenfant ventures to the far edge of the solar system, where he discovers a strange artifact left behind by an alien civilization: A gateway that functions as a kind of quantum transporter, allowing virtually instantaneous travel over the vast distances of interstellar space. What lies on the other side of the gateway? Malenfant decides to find out. Yet he will soon be faced with an impossible choice that will push him beyond terror, beyond sanity, beyond humanity itself. Meanwhile on Earth the Japanese scientist Nemoto fears her worst nightmares are coming true. Startling discoveries reveal that the Moon, Venus, even Mars once thrived with life?life that was snuffed out not just once but many times, in cycles of birth and destruction. And the next chilling cycle is set to begin again . . .

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What lies on the other side of the gateway? Malenfant decides to find out. Yet he will soon be faced with an impossible choice that will push him beyond terror, beyond sanity, beyond humanity itself.

Meanwhile on Earth the Japanese scientist Nemoto fears her worst nightmares are coming true. Startling discoveries reveal that the Moon, Venus, even Mars once thrived with life -- life that was snuffed out not just once but many times, in cycles of birth and destruction. And the next chilling cycle is set to begin again...

“Breathtaking in its originality and scope... [An] impressive parade of wonders.”
   THE WASHINGTON POST

Customer Reviews:

Slow but good.......2007-08-08

Manifold is a series of books with big, visionary concepts, and Space is no different. This time the twist on the Fermi paradox has the aliens existing and actually quite near the Earth. Reid Malenfant investigates with a mysterious Japanese scientist Nemoto. The first contact is made and the truth starts to unfurl...

As I said, the ideas are big - seriously big. The flow of the story isn't always fast enough, it all gets a bit too slow at times. Still, one has to admire Baxter's vision and while parts of the book were slightly boring, the whole of the story was definitely captivating enough to get me through the slower bits.

Manifold: Space offers an interesting what-if scenario of the future of humankind in a world that has extraterrestrial life.

Technically great, but ultimately cold and depressing.......2006-11-25

Manifold: Space is a very frustrating novel. As with its predecessor, Manifold: Time, it is brimming with great hard science surrounding a very good premise. In the Manifold series, Baxter gives detailed and extended perspectives to "Fermi's paradox". Fermi articulated that, due to the huge extent and age of the universe, either life on Earth is completely unique and we are alone in the universe (the basis of in Manifold: Time), or life must be everywhere and we simply have yet to discover it (as espoused in this novel). The paradox with the 2nd view is that if life is everywhere, the age of the universe implies that we cannot be the first cognitive, noise-making intelligence; why, then, haven't we found evidence of this other life? Baxter's answer to this paradox is quite interesting: he ties up the multiple story threads of Manifold: Space with a good ending.

Unfortunately, problems Baxter had with characterizations in Manifold: Time come to full flower in Manifold: Space. This novel is too seriously flawed too make it an overall enjoyable read.

There is a lack of any kind of human "community" in Manifold: Space. The various astronauts/space discoverers that are central to the story are all unhappy loners. They go into space alone, seemingly unaided, and appear to have no friends or colleagues. The first astronaut to set foot on Venus, as one example, never communicates with anyone beyond a single individual on earth, and that individual is also a loner. Baxter often beautifully elucidates the technical side of space exploration, but appears to have no clue as to the human elements. Manifold: Space is a cold unhappy story: interesting science populated by lonely, depressed individuals.

MANIFOLD ZZZZZZZZzzzzzzz.......2006-06-03

Let me start by saying I am an avid Sci-Fi reader, the longer the book the better. I believe a good story takes a long time to tell ala Harry Turtledove. So when I read the plot and saw the book size, I thought I was in for a good read. I could not have been more incorrect. Here are my bullet points:

1) Excellent plot idea, but I think Baxter forgot what is was after the first 12 pages. I have never been a part of such a plot trainwreck as this. Its like he compiled a dozen different stories into one without any of them going anywhere.
2) Character developement was horrendous. He needs to take a lesson from Harry Turtledove or Kevin Anderson. Those guys can present twenty or more characters, make you remember and love each one, this guy can't do it with two.
3) Science. About the only thing this guy can get across, but its not involving, its like reading a NASA technical paper, done with lots of coffee.
4) Writing. He needs to take a lesson from Jack McDevitt. You need to make the book an incredible journey independent of the ending. I begged for the ending just to finish the book. If Nemoto was one thousand years old, I felt like it half way through the book. Baxter's writing style is cold and easily forgotten.

The plot to this review, skip this book. It goes everywhere and nowhere all at once. I implore Baxter to work on his writing style and present more concise plot lines, eliminate all the extra about prehistoric men crapping on themselves on Jupiter's moon and stick to the plot. The editors need to quite screwing around and edit these books. Like previous readers, the book could have been cut in half...

Hope for our future.......2005-12-07

This is an odd story. It develops into a quest for the meaning of life. I enjoyed it's start: China and Japan have taken the lead in space exploration because the United States has not shown the desire to put money into such ventures. Then there is the time traveling aspect: going through portals to other systems takes the time it takes for light to travel, so rather then instantenous appearance, it is five or six years to get there and the same amount to get back. Rather simple, but no one has mentioned it before.
The ending is a treat, a bit unsatisfying because you want to see happiness and success, and it is there in a sense, it is not what you expect.

Manifold arghhhhhh!.......2005-10-10

I think I would really enjoy a physics class taught by Mr Baxter, since the man clearly knows his science, knows which sources to go to and is able to synthesize them in ways that he can explain them without your head exploding. Also, he's able to extrapolate those ideas into some interesting scenarios that suggest some fun possibilities and open the mind up for more speculation. Alas, he's not able to translate that yet into novels with real emotional heft and the bigger his ideas get, the more the story tends to leave you behind. This book is the follow-up to Manifold: Time and as the title implies, while this book still requires great leaps in time, this time out we're more about moving outward than jumping forward. Jumping off the conclusions set by the last novel, Baxter essentially hits the reboot button and drops us in a totally different universe, although some of the characters are still the same. The most notable is our hero Reid Malenfant, who is older than the first book but just as obsessed with mankind securing their destiny beyond the stars. Instead of wondering how we're going to survive, however, he's more interested in why they aren't any other intelligent races in the galaxy, or at least why it seems that way. Meanwhile, an intelligent race does apparently decide to move into the solar system from far away and that's where the fun starts. Always willing to boldly go (split infinitive and everything), Reid disappears into a portal that was left by some prior alien race and the voyage of discovery begins. As I mentioned before, Baxter is fiend when it comes to idea and everything is well thought out. From the weird robot-like Gajin that show up, to all the other aliens that pop in every so often, to the effects of time and history when extrapolated thousands upon thousands of years into the future, it's like he's taking you through an exhibit hall in his mind, showcasing all the fun science-based ideas he has and dressing them up as science-fiction. However, it doesn't take long before the reader quickly starts to lose the plot and everything begins to drown in a sea of glorious information. There are gates that allow people to travel but since it's lights-speed, time dilation still applies and when they come back to Earth, it's hundreds of years in the future each time, leading to effects that aren't much different from what the soldier suffered in Joe Haldemann's famous The Forever War, namely culture shock and the sense of dissociation and alienation. Things progress and Baxter shows us people on the moon, people on different planets, different aliens in other star systems, the galaxy, the picture keeps blowing up bigger and bigger until you almost can't contain it. But we don't really ever find out what it all means. The cover copy on the back of the book suggests that the main thrust of the plot has something to do with the reasons why intelligent life flares up and then dies out, leaving artifacts scattered all around the galaxy that other races pick up on, and Baxter sprinkles plenty of hints that the solar system was previously used by other races for various purposes, but it's buried under such a sea of static that it's hard to sort through it all. Especially since Baxter seems to be acting like an excited child, throwing each new idea in our face almost as he comes up with it, "You like this? Well how about this? Or this? Or even this?" until you're almost numb from it. By the time he gets around to answering the central question, almost five hundred pages later, you're been dazzled by as many marvels as science can handle but feeling strangely empty at the same time. It almost feels like "Who cares?" and regardless of anything else, I doubt that's the reaction he intended. Still, he gets an A for effort and the individual moments are extraordinary, showing that he's a mind possessed of a far vision and the fact that he's able to take all this knowledge and assemble it into something resembling a story is amazing in itself. But when you add it all up, it's not quite everything it's supposed to be. Hard science junkies will probably eat it up like the candy it is, everyone else looking for slightly more emotional content may find it rough going at times but for anyone who wants to see if all those complicated ideas that involve words with too many letters actually might mean something, well this is as good a place to start as any.

Mathematical Implications of Einstein-Weyl Causality (Lecture Notes in Physics)

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Mathematical Implications of Einstein-Weyl Causality (Lecture Notes in Physics)
Hans-Jürgen Borchers , and Rathindra Nath Sen
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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The present work is the first systematic attempt at answering the following fundamental question: what mathematical structures does Einstein-Weyl causality impose on a point-set that has no other previous structure defined on it? The authors propose an axiomatization of Einstein-Weyl causality (inspired by physics), and investigate the topological and uniform structures that it implies. Their final result is that a causal space is densely embedded in one that is locally a differentiable manifold. The mathematical level required of the reader is that of the graduate student in mathematical physics.

String Topology and Cyclic Homology (Advanced Courses in Mathematics - CRM Barcelona)

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String Topology and Cyclic Homology (Advanced Courses in Mathematics - CRM Barcelona)
Ralph L. Cohen , Kathryn Hess , and Alexander A. Voronov
Manufacturer: Birkhauser
ProductGroup: Book
Binding: Paperback

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ASIN: 3764321822

Book Description

The subject of this book is string topology, Hochschild and cyclic homology. The first part consists of an excellent exposition of various approaches to string topology and the Chas-Sullivan loop product. The second gives a complete and clear construction of an algebraic model for computing topological cyclic homology. The book provides many references for the reader wishing to learn more about the subject, to which it gives a perfect introduction. It is therefore suitable for both graduate students and established researchers. It is certainly the best source of much information that was until now available only to specialists and covers material from the elementary bases to the most recent developments.

Sobolev Spaces on Riemannian Manifolds (Lecture Notes in Mathematics)

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buy Hebey's newer book instead

Sobolev Spaces on Riemannian Manifolds (Lecture Notes in Mathematics)
Emmanuel Hebey
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540617221

Book Description

Several books deal with Sobolev spaces on open subsets of R (n), but none yet with Sobolev spaces on Riemannian manifolds, despite the fact that the theory of Sobolev spaces on Riemannian manifolds already goes back about 20 years. The book of Emmanuel Hebey will fill this gap, and become a necessary reading for all using Sobolev spaces on Riemannian manifolds.
Hebey's presentation is very detailed, and includes the most recent developments due mainly to the author himself and to Hebey-Vaugon. He makes numerous things more precise, and discusses the hypotheses to test whether they can be weakened, and also presents new results.

Customer Reviews:

buy Hebey's newer book instead.......2007-04-13

At the time that this book was written, it may have filled an important gap in the literature in that it dealt with Sobolov spaces on general Riemannian manifolds (even noncompact ones) as opposed to the usual treatments in euclidean space. However, since that time, there have been other, better books on the subject, most notably, Hebey's own "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities," published in 2000 by the AMS. The latter book includes the entire contents of this one (with many of the same chapter titles) plus a lot more, such as the Nirenberg problem, Nash's inequality, and manifolds with boundaries. In fact, the newer book has more than twice as many pages, and is a much, much nicer printing as well. I can't think of any reason to buy this Springer version, unless it was dramatically cheaper, which it's not. The only reason why I gave it 2 stars is because the book really isn't that bad in itself - 10 years ago I would've rated it higher.

Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Applied Mathematical Sciences)

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Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Applied Mathematical Sciences)
Stephen Wiggins , G. Haller , and I. Mezic
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 038794205X

Book Description

In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally hyperbolic invariant manifolds and foliations of normally hyperbolic invariant manifolds. In recent years these techniques have been used for the development of global perturbation methods, the study of resonance phenomena in coupled oscillators, geometric singular perturbation theory, and the study of bursting phenomena in biological oscillators. "Invariant manifold theorems" have become standard tools for applied mathematicians, physicists, engineers, and virtually anyone working on nonlinear problems from a geometric viewpoint. In this book, the author gives a self-contained development of these ideas as well as proofs of the main theorems along the lines of the seminal works of Fenichel. In general, the Fenichel theory is very valuable for many applications, but it is not easy for people to get into from existing literature. This book provides an excellent avenue to that. Wiggins also describes a variety of settings where these techniques can be used in applications.

Hyperbolic Manifolds and Discrete Groups

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Hyperbolic Manifolds and Discrete Groups
Michael Kapovich
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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Knots and Links

ASIN: 0817639047

Book Description

Hyperbolic Manifolds and Discrete Groups is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout is on the "Big Monster," i.e., on Thurston's hyperbolization theorem, which has not only completely changed the landscape of 3-dimensional topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book presents the first complete proof of Thurston's hyperbolization theorem in the "generic case" and an outline of Otal's proof of the hyperbolization theorem for manifolds fibered over the circle.

This important work contains an extended treatment of the theory of Kleinian groups and group actions on trees, including such key topics as:

* the Kazhdan--Margulis--Zassenhaus theorem

* the Klein and Maskit combination theorems

* the Mostow rigidity theorem

* the Douady--Earle extension theorem for homeomorphisms of the circle

* the smoothness theorem for representation varieties of Kleinian groups

* the Ahlfors finiteness theorem

* the Brooks deformation theorem

* characterization of pseudo-Anosov homeomorphisms

* compactification of character varieties via group actions on trees

The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds (Monographs and Textbooks in Pure and Applied Mathematics, Vol. 96) (Monographs ... in Pure and Applied Mathematics, Vol 96)

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An Excellent Prolegomena to Elementary Topology.
An explanation of how things can be twisted in space
A must-read
Topology at its entertaining and intuitive best.

The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds (Monographs and Textbooks in Pure and Applied Mathematics, Vol. 96) (Monographs ... in Pure and Applied Mathematics, Vol 96)
Jeffrey R. Weeks
Manufacturer: Dekker
ProductGroup: Book
Binding: Hardcover

Differential Topology: First Steps

ASIN: 082477437X

Book Description

Customer Reviews:

An Excellent Prolegomena to Elementary Topology........2001-03-26

I am trying to teach myself topology and find good elementary books in this subject rather slim in number. That is why I love this book. I can actually understand it. Weeks' clear prose and entertaining style make thinking about space an enjoyable activity in itself. I only wish a complete course could be written in this manner. Readers should not gain the impression that this even begins to scratch the surface of topology. It is more like an entertaining survey to motivate one into delving further into this wonderful subject.

By the way, I was suprised to realize that cosmologists have been confusing curvature with omega all this time. If something so fundamental can be mistaken how many other things in physics might be clearer with a sound topologic training?

An explanation of how things can be twisted in space.......2000-11-24

Any author attempting to explain and visualize dimensions higher than three and/or the elliptic and hyperbolic geometries is engaged in a significant undertaking. In this book, Weeks does succeed in doing both but the reader is presented with a difficult task.
With 141 exercises and plenty of illustrations packed into 324 pages, it is short on explanation and the reader is forced to learn by problem solving. This is not to say that the exercises are poorly developed. On the contrary, they and the illustrations are very well done. However, doing an exercise after every few paragraphs does make the book a slow read, and in many cases it is necessary to understand a problem before the next material can be comprehended. Fortunately, complete solutions to all problems are given at the end of the book, but even so, a great deal of thought must be given to some of them before they are understood. As the book progressed, I found myself reading only fifteen to twenty-five pages on any given day. This necessitated a great deal of back-pedaling to previous illustrations and exercises, but it was the limit that I seemed able to comprehend at any given setting.
Beginning with Flatland (by A Square-actually Edwin A. Abbott) and going through the creation of manifolds, the presentation of the basic concepts, like all of the text, is very well written. It is just unfortunate that there is not more of it. For example, in Chapter 9 (concerning spheres), there are seven exercises and five and one-half pages of illustrations packed in twelve pages. Chapter 17 (describing bundles), has thirteen problems and seven and one-half pages of diagrams in a total of fifteen pages. Illustrations are valuable, but in this case they describe abstract phenomena not easily followed, and more words than usual are needed to explain precisely what is occurring.
And so, in conclusion, this book is highly recommended for those who wish to learn about the properties of manifolds and surfaces and are highly motivated to do so. But lacking that, the chances are very good that you will not make it beyond the midpoint.

Published in Journal of Recreational Mathematics, reprinted with permission.

A must-read.......2000-05-13

As a high school student, many of the books regarding topology and manifolds are not always easy to follow. Jeffrey Weeks implements several illustrations and a simple yet poetic writing style to effectively portray worldy concepts to readers of all levels (even high scool students). It is obvious that he has a passion about his subject and has definitely inspired me to open my mind to mathematical ideas that had never found me before. I recommend this book to all readers!

Topology at its entertaining and intuitive best........1999-05-08

It's been many years since I read this book. But it was terrific, and I cannot praise it too highly. I have to believe that anyone from a bright junior higher to a scientific Nobel laureate, if not versed in the subject, can read this book with pleasure and profit. This book is light and easy to work through, but rich in content. It will carry you along with almost no effort while introducing you to exotic things from Klein bottles to periodic and curved spaces. The line diagrams filling the book are crystal clear and add charm all by themselves. This book could only be created by someone who loves the subject and loves to teach. All hats off to Jeffrey Weeks! -- no bones about it, he has produced a masterpiece of math education. I doubt that topology has ever been presented in a more winsome way.

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