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Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. With the minimum of prerequisites, Dr. Reid introduces the reader to the basic concepts of algebraic geometry, including: plane conics, cubics and the group law, affine and projective varieties, and nonsingularity and dimension. He stresses the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book contains numerous examples and exercises illustrating the theory. ... Read more

Customer Reviews (3)

Exceptionally good book; but make sure this is what you need
This book is intended to provide us with a short (135 pages), down to earth and fluently motivated introduction to algebraic geometry. And it does a great job. While the author does not clearly state his intentions in advance, I think it would be safe to assume that this is meant to accompany a more standard text on the subject (Hartshorne, Harris, Shafarevich, etc), and that the author's main goal was to give the quickest possible route to the heart of the subject, making sure the reader stays interested throughout rather than that he is presented with the firmest logical structure. I would like to stress that despite of what I wrote so far, this book does present rigorous proofs and clear definitions.

The style is friendly, straightforward and unpretentious. Everything is well motivated, and one occasionally gets to hear the author's personal perspective or view about a certain topic. I will quote two examples. When discussing the Zariski topology, the author writes "The Zariski topology may cause trouble to some students; since it is only being used as a language, and has almost no content, the difficulty is likely to be psychological rather than technical". This was very calming for me to read, as I have been previously struggling with the "deep meaning" of the Zariski topology, and no book has had the honesty to tell me that I shouldn't worry that much about it. As a second example of the author's style, after a Q.E.D. in page 53 the author explains that "The proof of (b) is a typical algebraist's proof: it's logically very neat, but almost completely hides the content: the real point is that ..."

Chapter 1 begins with the concrete example of conics, intended to motivate the later definitions of the projective plane. Next elliptic curves and their group law is discussed. The chapter ends with a brief survey of the genus of curves.

Chapter 2 is more technical; its purpose is to build the algebraic foundations needed for Hilbert's Nullstellensatz. Among topics covered are Noetherian rings, Hilbert's Basis Theorem, algebraic sets, the Zariski topology, prime ideals and a nice motivation for the Nullstellensatz. Next coordinate rings, morphisms, varieties and other standard topics are introduced.

Chapter 3, titled "Applications", uses the previous material to discuss some nice geometric topics. I especially enjoyed the section on the 27 lines on a cubic surface.

I would highly recommend this book to anyone not very familiar with algebraic geometry; for instance, it could be a good reading to decide if you want to take a more serious study (e.g. a university course) of the subject. If I were to suggest only one text for someone who just wants to know what algebraic geometry is all about, it would definitely be this one.

Unreadable but well-organized
It is difficult to see who this book is aimed at.Perhaps the extremely gifted undergraduate who can fill in sketchy, incomplete, difficult proofs, but has also taken courses?My professor (a topologist) even had a difficult time presenting the material as-is and solving the exercises, as very few examples were given, hence it was unclear exactly what was required for a satisfactory proof of the questions as stated.Reid, probably in an effort to save space, delegates difficult steps of proofs to the reader by declaring them "obvious," making the book practically unreadable to the average undergraduate student.The notation is used strangely and the typesetting is awkward.

The proof of the 27-lines theorem is interesting and a decent capstone for the introductory subject.However, I did not feel as though I had deepened my knowledge of algebraic geometry as a result, only having learned the bare minimum to approach one useless (albeit entertaining) theorem.

If you have to use this book I recommend buying another one to supplement the background knowledge and to figure out how to complete the proofs.

baked just right for the first timers !
There are many good books on the subject of algebraic geometry, so what was the use of one more - asks the author in the preface to this book.But there are none -at the UG level- which for the first time reveal to theyounger mathematicians the secrets of this vast and growing subject. Thebook treats every new concept with the rigour that keeps in mind the levelit is meant for, and yet maintains its mathematical "beauty" -setting firmly the basics for those who would want to take up this courseat an advanced level as well as keeping the more casual mathematics readerinterested. ... Read more

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This text presents an integrated development of core material from several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.

Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure theory and representation theory of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Milicic's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text.

There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert. ... Read more

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This classic work (first published in 1947), in three volumes, provides a lucid and rigorous account of the foundations of modern algebraic geometry.The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties but geometrical meaning has been emphasized throughout. This first volume is divided into two parts.The first is devoted to pure algebra: the basic notions, the theory of matrices over a non-commutative ground field and a study of algebraic equations.The second part is in n dimensions.It concludes with a purely algebraic account of collineations and correlations. ... Read more

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Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch-Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena. ... Read more

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Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text. ... Read more

Customer Reviews (7)

Excellent Book...very atypical for a math book, but I like it!
People tend to have strong feelings about this book.In my opinion, the people who dislike it are those who expect it to be like a typical graduate-level math book.This book is extremely atypical for a math book; it's not meant to be read linearly, and the topics in it do not follow a typical logical dependency.Personally, I find it to be outstanding; my only complaint about it is that I wish there were more books like it!

Commutative algebra and algebraic geometry are extremely difficult subjects requiring a great deal of background.This book is written as a sort of intermediary text between introductory abstract algebra books with a full and exposition of algebraic structures, and advanced, highly technical texts that can be difficult to follow and grasp on a technical level.As such, this book focuses on developing intuition, and discussing the history and motivation behind the various mathematical structures presented.It assumes that most of the other aspects of the subject, including both the elementary expositions, and the more advanced technical details, can be found elsewhere (although, believe me, this book certainly has its share of both elementary expositions and advanced technical details!)

I think this book is actually better for self-study than for use as a textbook.Most of the people I have known who have used it as a textbook have been frustrated with it.Either way, it needs to be supplemented by other books.Personally, on algebra, I like the Dummit and Foote, Isaacs, and Lang books.Those three books have very little overlap with each other, and very little overlap with this book, and they offer a very useful difference of perspectives where they do overlap!I also would recommend reading the more elementary book by Cox, Little, and O'Shea, which can help you get a feel for the subject of algebraic geometry.Many people see this book's primary purpose as preparation for Robin Hartshorne's "Algebraic Geometry".I can't say, however, how effective it is at that purpose, as no matter how far I get in this book, all but a few sections from that book still remain quite far beyond my grasp.

Not for the beginner
Well, the strength of this book lies in where it takes you. There is so much material here that when finished, you'll be prepared for a lot. Personally I think it is too wordy (my preferance is Atiyah & MacDonald) and the typesetting overall isn't all that impressive, so read up or consult other texts before/during your first encounter. M.Reids book is a better place to start.

Good book of reference
I purchased this as a book of reference.When I want to know something about Commutative Algebra (while reading Hartshorne's Algebraic Geometrry), I like a standard book of reference.But it seems a good book to learn commutative algebra aswell.

very good, but should be read slowly
Some proofs are somewhat abstract to the beginner. Although you are forced to check them on the paper, I think it is very good for the study. Also, you need a professor to instruct you, because in math, any language could only express the part of the oringins. Anyway, algebraic geometry is the course that you have to have a good professor to help you, otherwise stop study this field. In one word, it is a very very good book, so read it slowly!!!!!!

Superb
If one is interested in taking on a thorough study of algebraic geometry, this book is a perfect starting point. The writing is excellent, and the student will find many exercises that illustrate and extend the results in each chapter. Readers are expected to have an undergraduate background in algebra, and maybe some analysis and elementary notions from differential geometry. Space does not permit a thorough review here, so just a brief summary of the places where the author has done an exceptional job of explaining or motivating a particular concept:

(1) The history of commutative algebra and its connection with algebraic geometry, for example the origin of the concept of an "ideal" of a ring as generalizing unique factorization.

(2) The discussion of the concept of localization, especially its origins in geometry. A zero dimensional ring (collection of "points") is a ring whose primes are all maximal, as expected.

(3) The theory of prime decomposition as a generalization of unique prime factorization. Primary decomposition is given a nice geometric interpretation in the book.

(4) Five different proofs of the Nullstellensatz discussed, giving the reader good insight on this important result.

(5) The geometric interpretation of an associated graded ring corresponding to the exceptional set in the blowup algebra.

(6) The notion of flatness of a module as a continuity of fibers and a test for this using the Tor functor.

(7) The characterization of Hensel's lemma as a version of Newton's method for solving equations. The geometric interpretation of the completion as representing the properties of a variety in neighborhoods smaller than Zariski open neighborhoods.

(8) The characterization of dimension using the Hilbert polynomial.

(9) The fiber dimension and the proof of its upper semicontinuity.

(10) The discussion of Grobner bases and flat families. Nice examples are given of a flat family connecting a finite set of ideals to their initial ideals.

(11) Computer algebra projects for the reader using the software packages CoCoA and Macaulay.

(12) The theory of differentials in algebraic geometry as a generalization of what is done in differential geometry.

(13) The discussion of how to construct complexes using tensor products and mapping cones in order to study the Koszul complex.

(14) The connection of the Koszul complex to the cotangent bundle of projective space.

(15) The geometric interpretation of the Cohen-Macauley property as a map to a regular variety. ... Read more

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This work serves as an introduction to the theory of projective geometry. Techniques and concepts of modern algebra are presented for their role in the study of projective geometry. Topics covered include: affine and projective planes; homogeneous co-ordinates; and Desargues' theorem. ... Read more

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Now back in print, this highly regarded book has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces, which include moduli spaces in positive characteristic, moduli spaces of principal bundles and of complexes, Hilbert schemes of points on surfaces, derived categories of coherent sheaves, and moduli spaces of sheaves on Calabi-Yau threefolds. The authors review changes in the field since the publication of the original edition in 1997 and point the reader towards further literature. References have been brought up to date and errors removed. Developed from the authors' lectures, this book is ideal as a text for graduate students as well as a valuable resource for any mathematician with a background in algebraic geometry who wants to learn more about Grothendieck's approach. ... Read more

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The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry. ... Read more

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The second volume of the Geometry of Algebraic Curves is devoted to the foundations of the theory of moduli of algebraic curves. Its authors are research mathematicians who have actively participated in the development of the Geometry of Algebraic Curves. The subject is an extremely fertile and active one, both within the mathematical community and at the interface with the theoretical physics community. The approach is unique in its blending of algebro-geometric, complex analytic and topological/combinatorial methods. It treats important topics such as Teichmüller theory, the cellular decomposition of moduli and its consequences and the Witten conjecture. The careful and comprehensive presentation of the material is of value to students who wish to learn the subject and to experts as a reference source.

The first volume appeared 1985 as vol. 267 of the same series.

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This is the first of three volumes on algebraic geometry. The second volume, Algebraic Geometry 2: Sheaves and Cohomology, is available from the AMS as Volume 197 in the Translations of Mathematical Monographs series.

Early in the 20th century, algebraic geometry underwent a significant overhaul, as mathematicians, notably Zariski, introduced a much stronger emphasis on algebra and rigor into the subject. This was followed by another fundamental change in the 1960s with Grothendieck's introduction of schemes. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject.

The book begins with a description of the standard theory of algebraic varieties. Then, sheaves are introduced and studied, using as few prerequisites as possible. Once sheaf theory has been well understood, the next step is to see that an affine scheme can be defined in terms of a sheaf over the prime spectrum of a ring. By studying algebraic varieties over a field, Ueno demonstrates how the notion of schemes is necessary in algebraic geometry.

This first volume gives a definition of schemes and describes some of their elementary properties. It is then possible, with only a little additional work, to discover their usefulness. Further properties of schemes will be discussed in the second volume. ... Read more

Customer Reviews (2)

Not good by comparison
Algebraic geometry in modern times is not an easy subject. A better introductory text is Introduction to Algebraic Geometry.
My beef with this type of Mathematical writing is old: the writer gives the student nothing but a new axiomatic language without "context" or contact with objective reality ( no real pictures of the geometry involved).
He also expects after we have read this badly written text to buy volumes 2 and 3? He is not alone in going over students heads in Algebraic Geometry.
I bought this hoping that it would give a decent introduction to Schemes.
It doesn't even give an a good introduction to Zariski topology or why
Zariski(T0) instead of Hausdorff (T2) ... ? The examples, problems anddefinitions are pretty bad too. If you want your grad students in massive depression while taking your course, use this as a text?I bought this book after doing several weeks of searching for a cheap book
that covered the areas I wanted to learn.
I've pretty much come to the conclusion there are some very strange people in this field and very few real teachers?
If in you are presenting a subject in Mathematics in an Axiomatic form like this, you have to tell the people why the axioms/ theorems are as they are:not just give definition in strange symbolsand prove using the same new notation.
I've seen worse than this text, but not by much?An Introduction to Homological Algebra for example.
Presenting Zariski tangent space without a diffeomorphism definition
is just really bad Mathematics with no excuse in my mind?
Presenting Schemes without reference to Galois theory is not a very good idea either? Not mentioning that Algebraic geometry uses Zariski topology because it excludes the transcendental numbers ( no algebraic variety has root that is Pi or e). Some bridge to measure theory for Schemes
seems necessary, since the use of "spectrum" in the definition tends to confuse the student for other areas that are more concretely defined?
The father of algebraic geometry is Descartes,yet he seems to never be mentioned. Instead Grothendieck appears everywhere where things get most dense? I repeat, if you are approaching a subject axiomatically, you have to made plain the basis for those axioms. And algebra without algebra ( polynomials)and geometry without geometry ( pictures)is probably very confusing to most students.

A Good Book Overall
A nice book with details worked out but quite a few typos. ... Read more

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This two volume work on ... Read more

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Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities. ... Read more

Customer Reviews (1)

Excellent
Statistical learning theory is now a well-established subject, and has found practical use in artificial intelligence as well as a framework for studying computational learning theory. There are many fine books on the subject, but this one studies it from the standpoint of algebraic geometry, a field which decades ago was deemed too esoteric for use in the real world but is now embedded in myriads of applications. More specifically, the author uses the resolution of singularities theorem from real algebraic geometry to study statistical learning theory when the parameter space is highly singular. The clarity of the book is outstanding and it should be of great interest to anyone who wants to study not only statistical learning theory but is also interested in yet another application of algebraic geometry. Readers will need preparation in real and functional analysis, and some good background in algebraic geometry, but not necessarily at the level of modern approaches to the subject. In fact, the author does not use algebraic geometry over algebraically closed fields (only over the field of real numbers), and so readers do not need to approach this book with the heavy machinery that is characteristic of most contemporary texts and monographs on algebraic geometry. The author devotes some space in the book for a review of the needed algebraic geometry.

Also reviewed in the initial sections of the book are the concepts from statistical learning theory, including the very important method of comparing two probability density functions: the Kullback-Leibler distance (called relative entropy in the physics literature). The reader will have to have a good understanding of functional analysis to follow the discussion, being able to appreciate for example the difference between convergence in different norms on function space. From a theoretical standpoint, learning can be different in different norms, a fact that becomes readily apparent throughout the book (from a practical standpoint however, it is difficult to distinguish between norms, due to the finiteness of all data sets). Of particular importance in early discussion is the need for "singular" statistical learning theory, which as the author shows, boils down to finding a mathematical formalism that can cope with learning problems where the Fisher information matrix is not positive definite (in this case there is no guarantee that unbiased estimators will be available). This is where (real) algebraic geometry comes in, for it allows the removal of the singularities in parameter space by recursively using "blow-up" (birational) maps. The author lists several examples of singular theories, such as hidden Markov models, Boltzmann machines, and Bayesian networks. The author also shows to generalize some of the standard constructions in "ordinary" or "regular" statistical learning to the case of singular theories, such as the Akaike information criterion and Bayes information criterion. Some of the definitions he makes are somewhat different than what some readers are used to, such as the notion of stochastic complexity. In this book it is defined merely as the negative logarithm of the `evidence', whereas in information theory it is a measure of the code length of a sequence of data relative to a family of models. The methods for calculating the stochastic complexity in both cases are similar of course.

In singular theories, one must deal with such things as the divergence of the maximum likelihood estimator and the failure of asymptotic normality. The author shows how to deal with these situations after the singularities are resolved, and he gives a convincing argument as to why his strategies are generic enough to cover situations where the set of singular parameters, i.e. the set where the Fisher information matrix is degenerate, has measure zero. In this case, he correctly points out that one still needs to know if the true parameter is contained in the singular set, and this entails dealing with "non-generic" situations using hypothesis testing, etc.

Examples of singular learning machines are given towards the end of the book, one of these being a hidden Markov model, while another deals with a multilayer perceptron. The latter example is very important since the slowness in learning in multilayer perceptrons is widely encountered in practice (largely dependent on the training samples). The author shows how this is related to the singularities in the parameter space from which the learning is sampled, even when the true distribution is outside of the parametric model, where the collection of parameters is finite. This example leads credence to the motto that "singularities affect learning" and the author goes on further to show to what extent this is a "universal" phenomenon. By this he means that having only a "small" number of training samples will bring out the complexity of the singular parameter space; increasing the number of training samples brings out the simplicity of the singular parameter space. He concludes from this that the singularities make the learning curve smaller than any nonsingular learning machine. Most interestingly, he speculates that "brain-like systems utilize the effect of singularities in the real world."
... Read more

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How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

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The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.

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In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Customer Reviews (19)

Terrible textbook
This book is horrible if regarded as a mathematics book. Like previous reviewers I feel there is a total lack of clarity and rigor. Definitions are lacking, perhaps the author feels it is better to provide a "intuitive" feel for the material, than just definingg things. He fails miserably. The fact that what we are really dealing with in this subject are functors((co)homology, homotopy ) is nearly absent from the text. Instead drawings and pictures that are meant to provide "geometric" feel are supplanted.
I would state that this book attempts to teach how to compute in and use the theory than have you understand how the theory is built. It is a book for using the oven, not understanding how it works.

More Hand-Waving Than an Orchestral Conductor
In the TV series "Babylon 5" the Minbari had a saying: "Faith manages."If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art.

To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics.Some folks really like that kind of stuff, I guess (judging from other reviews).Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts.When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad:inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date.

Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route with pen and paper.I concede this, but only to a point.The gaps here are so numerous that, to fill them all in, a reader would be spending a couple of days on each page of prose.It is not realistic.Some have charged that this text reads like a pop science book, while others have said it is extremely difficult.Both charges are true.Never have I encountered such rigorous beer-coaster explanations of mathematical concepts.Yet this book seems to get a free ride with many reviewers, I think because it is offered for free.In the final analysis is it a good book or a bad book?Well, it depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory.

First, one year of graduate algebra is not enough, you should take two. Otherwise while you may be able to fool yourself and even your professor into thinking you know what the hell is going on, you won't really.Not right away.Ignore this admonishment only if you enjoy applying chaos theory to your learning regimen.

Second, you better have a well-stocked library nearby, because as others have observed Hatcher rarely descends from his cloud city of lens spaces, mind-boggling torus knots and pathological horned spheres to answer the prayers of mortals to provide clear definitions of the terms he is using.Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion.

Third: yes, the diagram is commutative.Believe it.It just is.Hatcher will not explain why, so make the best of it by turning it into a drinking game.The more shots you take, the easier things are to accept.

In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help).It flummoxed me for a long while.The books I learned my point-set topology and modern algebra from did not prepare me for this "expanded" use of the notation usually reserved for quotient groups and the like.Munkres does not use it.Massey does not use it.No other topology text I got my hands on uses it.How did I figure it out?Wikipedia.Now that's just sad.Like I said earlier:one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time.

Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts.Many are pathological, being either extremely convoluted or torturously long-winded -- they usually can be safely skipped.

One specific gripe. The development of the Mayer-Vietoris sequence in homology is shoddy.It's then followed by Example 2.46, which is trivial and uncovers nothing new, and then Example 2.47, which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles." Oh really?(Cue clapping back-up chorus: "Well, ya gotta have faith...")That's the end of the "useful" examples at the Church of Hatcher on this important topic.

Another gripe. The write-up for delta-complexes is absolutely abominable. There is not a SINGLE EXAMPLE illustrating a delta-complex structure.No, the pictures on p. 102 don't cut it -- I'm talking about the definition as given at the bottom of p. 103.A delta-complex is a collection of maps, but never once is this idea explicitly developed.

A final gripe.The definition of the suspension of a map...?Anyone?Lip service is paid on page 9, but an explicit definition isn't actually in evidence.I have no bloody idea what "the quotient map of fx1" is supposed to mean. I can make a good guess, but it would only be a guess. Here's an idea for the 2nd edition, Allen: Sf([x,t]) := [f(x),t]. This is called an explicit definition, and if it had been included in the text it would have saved me half an hour of aggravation that, once again, only ended with Wikipedia.

But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.27 -- singular and simplicial homology groups of delta-complexes are isomorphic), I have to grant that Hatcher does leave just enough breadcrumbs to enable me to figure things out on my own if given enough time.I took one course that used this text and it was hell, but now I'm studying it on my own at a more leisurely pace.It's so worn from use it's falling apart.Another good thing about the book is that it doesn't muck up the gears with pervasive category theory, which in my opinion serves no use whatsoever at this level (and I swear it seems many books cram ad hoc category crapola into their treatments just for the sake of looking cool and sophisticated).My recommendation for a 2nd edition:throw out half of the "additional topics" and for the core material increase attention to detail by 50%.Oh, and rewrite Chapter 0 entirely. Less geometry, more algebra.

Really bad as a "readable" texbookbut good reference
I am not able to understad why people seems to love this book my feelings, beeing mixed, are perhaps closer to hate.

The book is OK if (and only if) you previously know the matter but the lack of clear definitions, the excessive reliance in reader geometrical intuition, the conversational style of demos the long paragraphs describing obscure geometric objects, etc make it very difficult to follow if it is your first approach to AT.

On the other hand has useful insigths if you already know the matter.

If the purpouse of the author has really been to write a "readable" book (as he told us repeatedly) I think the attemp is a complete failure.

On the other handthe "Table of contents" is excellent and is a very good book for teachers,I think this is the reason of itspopularity.

If you can afford the cost, purchase J Rotman "An introduction to Algebraic Topology" and you really will get a "readable" book

amazing book, but caveat emptor
I think that Allen Hatcher has given us all something very valuable in this book.If you are like me, you've had those moments when reading in a math book when you read a sentence, and your eyes shoot open and you suddenly feel like someone has been standing behind you that you never knew was there.There are lots of those kinds of sentences in this book.On the other hand, I view it as a supplement to a book like Munkres or Bredon that provides the rigor necessary to allow the learner to figure out the topologist's geometric language.I have used these three and found them to compliment one another well.

excellent modern introduction
This is an excellent introduction to the subject. It's affordable, well-written, and the topics are well chosen. The presentation is modern, but includes enough intuition that the fairly naive reader (e.g., me) can see the point of things. I needed to (re)learn topology for a research project I was part of in the intersection of math/CS/statistics and this book was a big help. I wish that he had included simplicial sets in the topics, because I like the way he writes and would like to have a more elementary exposition tied to the rest of the book (I eventually found an expository paper that did a pretty good job, but worked out examples would still help with that topic), but it can't include everything. I highly recommend this book to anyone trying to get started in this fascinating subject. It will just scratch the surface, but it does a good job of that. ... Read more

Product Description
Here is an introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences. The book is well illustrated and contains several hundred worked examples and exercises. From the familiar lines and conics of elementary geometry the reader proceeds to general curves in the real affine plane, with excursions to more general fields to illustrate applications, such as number theory. By adding points at infinity the affine plane is extended to the projective plane, yielding a natural setting for curves and providing a flood of illumination into the underlying geometry. A minimal amount of algebra leads to the famous theorem of Bezout, while the ideas of linear systems are used to discuss the classical group structure on the cubic. ... Read more

Customer Reviews (1)

Does what it claims
We used this for a class introducing algebraic geometry.It's necessary to know linear algebra and multivariable calculus, and very helpful to know groups, rings, and fields.The book keeps it's feet on the ground (equal distribution of computation and abstraction in problems).Some examples throughout.Very good as an introduction. ... Read more

Product Description
These volumes are based on lecture courses and seminars given at the LMS Durham Symposium on the geometry of low-dimensional manifolds.This area has been one of intense research recently, with major breakthroughs that have illuminated the way a number of different subjects (topology, differential and algebraic geometry and mathematical physics) interact. ... Read more

Product Description
In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. ... Read more