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The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem--the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most relevant contributions to the subject in the years 1959 to 1974. These include the pinching (or sphere) theorem, Berger's theorem for symmetric spaces, the differentiable sphere theorem, the structure of complete manifolds of non-negative curvature, and finally, results about the structure of complete manifolds of non-positive curvature. Emphasis is given to the phenomenon of rigidity, namely, the fact that although the conclusions which hold under the assumption of some strict inequality on curvature can fail when the strict inequality on curvature can fail when the strict inequality is relaxed to a weak one, the failure can happen only in a restricted way, which can usually be classified up to isometry. Much of the material, particularly the last four chapters, was essentially state-of-the-art when the book first appeared in 1975. Since then, the subject has exploded, but the material covered in the book still represents an essential prerequisite for anyone who wants to work in the field. ... Read more

Customer Reviews (2)

The basic reference
I am of course partial to the book ,I was graduate student at SUNY Stony Brook.
Still,I think this is the basic reference in the subject , a book that I like
to go back to review the most important theorems.
Clear, consise ,well written.
It is quite impissible to work in Riemanian Geometry without "Cheeger Ebin"

all you ever wanted to know about differential geometry
But thought the book was out of print. This classic book was published by North Holland in the late seventies, was photocopied by every grad student in geometry, and finally has been reprinted by the American Math Society. The book's goal (achieved) is to get you up to speed and working as quickly as possible. Riemannian geometry is covered from scratch (a la Milnor's Morse Theory (Annals of Mathematic Studies AM-51) but they don't stop there, and prove all of the basic comparison results. ... Read more

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"18 THEOREMS OF GEOMETRY is the precise book of choice for students preparing for the SAT, GED and the London Exams. Incidentally geometry happens to be a requirement in these and most other public exams today. Therefore we trust that like many who have benefited from tutoring offered by its author, you too will take advantage of the book that he produces.
In a few pages you have in your hand the book that guarantees your easy success in the exam that you are preparing for. The book begins by stating each theorem in a concise modern English followed by a mathematical proof and examples drawn from past exam papers.Together with clear diagrams, this approach is sure to engrain the concepts in the students to be able to come up with a solution when challenged with a problem in an exam.
"
... Read more

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College Geometry offers readers a deep understanding of the basic results in planegeometry and how they are used. Its unique coverage helps readers master Euclidean geometry, in preparation for non- Euclidean geometry. Focus on plane Euclidean geometry, reviewing high school level geometry and coverage of more advanced topicsequips readers with a thorough understanding of Euclidean geometry, needed in order to understand non-Euclideangeometry. Coverage of Spherical Geometry in preparation for introduction of non-Euclidean geometry. A strongemphasis on proofs is provided, presented in various levels of difficulty and phrased in the manner of present-daymathematicians, helping the reader to focus more on learning to do proofs by keeping the material less abstract. For readers pursuing a career in mathematics. ... Read more

Customer Reviews (2)

worst geometry book EVER.
This is the worst geometry book available.There aren't any extra examples for clarification or more practice and the explanations are terrible.I DO NOT RECOMMEND THIS BOOK TO ANYONE.

Geometry: Theorems and Constructions
This was the text book we used in a geometry class at Western Carolina University.The book goes over basic high school concepts to more advanced levels with clear examples with some figures shown.The reason for buying my own copy was to keep as a reference guide in teaching and tutoring geometry (high school and college levels). ... Read more

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This book is intended for amateurs, students and teachers.The author presents partial results which could be obtained with exclusively elementary methods.The proofs are given in detail, with minimal prerequisites.An original feature are the ten interludes, devoted to important topics of elementary number theory, thus making the reading of this book self-contained.Their interest goes beyond Fermat's theorem. The Epilogue is a serious attempt to render accessible the strategy of the recent proof of Fermat's last theorem, a great mathematical feat. ... Read more

Customer Reviews (5)

Title is misleading but a pretty good book
First, I give this book for 4 stars because the title is misleading. The background needed to understand this book is a major in mathematics at the undergrad level. If you were a math major and earned your degree several years ago, you may still have trouble following the book. It is for "amateurs" with a math degree and who are still "in touch" with the math they learnt. The book is however quite good; give a good historical narrative along with enough mathematics to satisfy the reader.

The problem is mathematics has gotten so much abstract and complicated during the last 50 years that it is impossible for someone not trained in the exact specialized field to follow what is going on. Since Fermat's last theorem has caught the public attention, people want to know what the fuss is all about. Alas, they really cannot understand it or even appreciate the mathematics in general without strong background in number theory, Galois theory, elliptic curves, and so on. All books that try to cater to the layman have to decide where to draw the line. If they water down the mathematics, then they really cannot explain how the proof was got satisfactorily; and if they throw in one too many equations, it becomes incomprehensible to many--even mathematicians not in that specialized field. My advice to the general public is to watch the video on Fermat's last theorem. I think for the layman visual media is better than a book. Just Google the excellent UKTV documentary on Fermat's last theorem.

Difficult book but great topic coverage
Solid coverage of proofs relating to Fermat's Last Theorem up to Kummer's Theory.You will find proofs for n=2, n=3, n=4, n=5, and n=7.Requires solid background in Algebraic Number Theory.For example, you should already have a good understanding of the Quadratic Law of Reciprocity, Quadratic Fields, and Congruences.If you don't, I recommend Elementary Number Theory for Congruences and the Quadratic Law of Reciprocity and Stark's An Introduction to Number Theory for Quadratic Fields.I would also recommend starting out with Edward's Book on Fermat's Last Theorem which includes detailed coverage of Kummer's Theory.

Great selection of material, difficult book
I find that this is a great book if you are an instructor or have a solid background in algebraic number theory.If you are unfamiliar with the Legendre Symbol, Gaussian integers, or the Law of Quadratic Reciprocity, you may wish to start out with a book such as Elementary Number Theory.If your are familiar with Algebraic Number Theory and wish to study in detail the Fermat Last Theorem proofs up to Kummer's Theory, this is a great book.I would recommend starting out with Edward's Book (Fermat's Last Theorem), for analysis of Euclid's proof of N=3.I found this very useful as an example of applications of Gaussian integers and Eisenstein integers.Ribenboim is one of the top experts about Fermat's Last Theorem and he is to praised for putting these beautiful proofs down.Even so, I would recommend purchasing other books to help explain this one.I found Stark's book very helpful in understanding Quadratic Fields.

"Amateur" mathematicians, that is !
If, like me, you were fascinated to hear that Fermat's so-called "last theorem" had been proven in 1995, then read Simon Singh or Amir Aczel's books popularizing the proof in outline, you probably wanted something more.

If, like me, you are a person who took some math in college, enjoys recreational mathematics books of the Douglas Hofstadter and Ian Stewart genre, and even sometimes picks up the odd number-theory book, you might consider yourself an "amateur."

If...if... this might seem like the book for you. I'd suggest that its not.

The mathematics in this book and its level of presentation was simply impenetrable by me. Not slow going... "no" going. That's frustrating to admit, but in a way fine, since it affirms of my admiration at a distance of the work that professional mathematicians do. I have seen many cited who state that Wiles' proof is simply beyond the ken of even 95% of working mathematicians. I believe this book must really be intended to serve some fraction of that group.Perhaps within the fold of mathematics these would consider themselves "amateurs". My two stars are offered only for them.

The book is simply not for the "lay" amateur. And Ribenboim's titling of it suggests that he does not even know that this lower caste, containing those of us who enjoy recreational mathematics and would describe ourselves as "amateurs", even exists. We know we exist as something mathematically distinct from the general population by the simple fact of the universally raised eyebrows that confront any mention of our interest in mathematics. Nevertheless, like any other species in a niche, we will have to continue to feed on a sparse supply of intellectual sustenance and learn to avoid the over-rich and indigestible fare of the higher forms.

Finally, if you haven't read Singh or Aczel I'd offer the former 5 stars and the latter 3 but recommend both. A truly fascinating story.

An excellant work, good for any serious study of FLT.
I am a math instructor and graduate student at PVAMU, and am working on a thesis detailing the history of attempts to prove and the Wiles proof of FLT. The text was easily readable and the proofs were very well done. I wasable to follow the logic and math of all the presented proofs very well. ... Read more

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Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague.
In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed. ... Read more

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This is one of the first monographs to deal with the metrictheory of spatial mappings and incorporates results in the theory ofquasi-conformal, quasi-isometric and other mappings.
The main subject is the study of the stability problem in Liouville'stheorem on conformal mappings in space, which is representative of anumber of problems on stability for transformation classes. To enablethis investigation a wide range of mathematical tools has beendeveloped which incorporate the calculus of variation, estimates fordifferential operators like Korn inequalities, properties of functionswith bounded mean oscillation, etc.
Results obtained by others researching similar topics are mentioned,and a survey is given of publications treating relevant questions orinvolving the technique proposed.
This volume will be of great value to graduate students andresearchers interested in geometric function theory.
... Read more

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This book is a translation of Professor Wu's seminal Chinese book of 1984 on Automated Geometric Theorem Proving. The translation was done by his former student Dongming Wang jointly with Xiaofan Jin so that authenticity is guaranteed. Meanwhile, automated geometric theorem proving based on Wu's method of characteristic sets has become one of the fundamental, practically successful, methods in this area that has drastically enhanced the scope of what is computationally tractable in automated theorem proving. This book is a source book for students and researchers who want to study both the intuitive first ideas behind the method and the formal details together with many examples. ... Read more

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This book begins by introducing the area method, and recent results in automating the area method. It can either be used as a geometry text for students and geometers, or be regarded as a monograph on machine proofs in geometry. By automating the area method, this book presents a systematic way of proving geometry theorems using traditional methods. The authors aim to make learning and teaching geometry easier through this book. ... Read more

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This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

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Provides a thorough account of many topics of geometry of numbers including lattices, reduction, Minkowski's Theorem, distance functions, packings, and automorphs. Includes excellent bibliographical references. Paper. ... Read more

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A collection of essays centered around mathematical mechanization, dealing with mathematics in an algorithmic and constructive manner, with the aim of developing mechanical, automated reasoning. Discusses historical developments, underlying principles, and features applications and examples. ... Read more

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Students will find this text useful for these reasons: 1) learning each rule, formula and principle. 2) learning each set of solved problems. 3) learning each set of supplementary problems. 4) integrating the learning of plane geometry. 5) Learning geometry through self-study. 6) extending plane geometry into solid geometry. CONTENTS: 1) Lines, angles and triangles 2) Methods of Proof 3) Congruent Triangles 4) Parallel Lines, Distances and Angle Sums 5) Parallelograms, Trapezoids, Medians and Midpoints 6) Circles 7) Similarity 8) Trigonometry 9) Areas 10) Regular Polygons and the Circle 11) Locus 12) Coordinate Geometry 13) Inequalities and Indirect Reasoning 14) Improvement of Reasoning 15) Constructions 16) Proofs of Required Theorems 17) Extending Plane Geometry into Solid Geometry. Also includes formulas for reference; table of trigonometric functions; table of squares and square roots; answers to supplementary problems. ... Read more

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This new, completely revised edition of a classic text introduces all elements necessary for understanding The Proof (Title of a PBS series dedicated to the proof of Fermat's Last Theorem) as well as new development and unsolved problems. Written by two distinguished mathematicians, Ian Stewart and David Tall, this book weaves together the historical development of the subject with a presentation of mathematical techniques. The result is a solid introduction to one of the most active research areas of mathematics for serious math buffs and a textbook accessible to undergraduates. ... Read more

Customer Reviews (7)

Not bad, but much to be improved.
I entirely agree with the review by Mr T. Luo.In the parts I and II, there exist many logical gaps in the exposition that require a substantial amount of effort to fill in.If this book is used as a textbook in a class, that may prove pedagogically benefiting.But self-studying newcomers to the subject will find the textbook hard to follow.I must add that there are many typos concerning fraktur, especially in chapter 5, which makes the reading frustrating.

Great Introductory Book to Algebraic Number Theory
I wasn't lucky enough to have the opportunity to have a class in algebraic number theory in college or graduate school, so I had to learn it on my own. This book was recommended to me by my friend Paul Pollack (author of Not Always Buried Deep) and the suggestion was fantastic, as I was able to learn algebraic number theory.

The book is written very clearly, it has nice exercises that make the theorems clearer and it covers the basic concepts from algebraic number theory.

This a great book to learn the basics of the subject.

skips too much
I guess the previous reviewers didn't try any of the exercises in the book. They are very good problems but the text is far from sufficient for us to solve the problems. For example, there is only one example in chapter 2 on how to find integral basis and it is a quadratic field. However, the 4th problem of this chapter is to find the discriminant of a degree-4 extension! At least the author should supply more theorems on integral basis so that we know how to start such a problem.
I feel like the author is very "Rudin" in his writing, neglecting all the details. Sometimes it's fun to fill in the details myself, but sometimes it can be rather annoying. I think a undergraduate textbook shouldn't skip too many steps in the proofs.

tough problems => good for the student
The motivation of explaining Fermat's Last Theorem is a nice device by which Stewart takes you on a tour of algebraic number theory. Things like rings of integers, Abelian groups, Minkowski's Theorem and Kummer's Theorem arise fluidly and naturally out of the presentation.

The inclusion of problem sets in each chapter also enlivens its appeal to a student. Typically, the first problems in each set are easy. But later problems can be quite formidable, and really give a good mental workout of the salient issues just covered in the chapter.

Very clear introduction to Algebraic Number Theory
This book is a very clear intoductino to ANT.It is a good first step for many reasons.One: it stays with algebraic number fields that are extensions of Q, the rational numbers.You get a good feel for the subject.When you go to more advanced books Q is replaced by other fields (P-adic, function fields, finite fields,..).
Two: He assumes very little and writes very clearly
Three: You only needs to read his Galois theory book for the prerequisite
Four: His book is what is usually left for the reader to do as an excersize in more advanced books. ... Read more

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Noncommutative Geometry and Cayley-smooth Orders explains the theory of Cayley-smooth orders in central simple algebras over function fields of varieties. In particular, the book describes the étale local structure of such orders as well as their central singularities and finite dimensional representations.

After an introduction to partial desingularizations of commutative singularities from noncommutative algebras, the book presents the invariant theoretic description of orders and their centers. It proceeds to introduce étale topology and its use in noncommutative algebra as well as to collect the necessary material on representations of quivers. The subsequent chapters explain the étale local structure of a Cayley-smooth order in a semisimple representation, classify the associated central singularity to smooth equivalence, describe the nullcone of these marked quiver representations, and relate them to the study of all isomorphism classes of n-dimensional representations of a Cayley-smooth order. The final chapters study Quillen-smooth algebras via their finite dimensional representations.

Noncommutative Geometry and Cayley-smooth Orders provides a gentle introduction to one of mathematics' and physics' hottest topics. ... Read more

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A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. They are scattered in research papers or outlined in surveys, and they often use topological notions not commonly known among combinatorialists or computer scientists.

This bookis the first textbook treatment of a significant part of such results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level (for example, homology theory and homotopy groups are completely avoided). No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

At the same time, many substantial combinatorial results are covered, sometimes with some of the most important results, such as Kneser's conjecture, showing them from various points of view.

The history of the presented material, references, related results, and more advanced methods are surveyed in separate subsections. The text is accompanied by numerous exercises, of varying difficulty. Many of the exercises actually outline additional results that did not fit in the main text. The book is richly illustrated, and it has a detailed index and an extensive bibliography.

This text started with a one-semester graduate course the author taught in fall 1993 in Prague. The transcripts of the lectures by the participants served as a basis of the first version. Some years later, a course partially based on that text was taught by G\"unter M. Ziegler in Berlin, who made book is based on a thoroughly rewritten version prepared during a pre-doctoral course I taught at the ETH Zurich in fall 2001.

Most of the material was covered in the course: Chapter 1 was assigned as an introductory reading text, and the other chapters were presented in approximately 30 hours of teaching (by 45 minutes), with some omissions throughout and with only a sketchy presentation of the last chapter. ... Read more

Customer Reviews (1)

Insight Pedagogy and Honesty
While allowing you to learn new concepts and definitions in a classical sense , this book shows what mathematics is about. It makes connections and gives you insight.
If you think that mathematics is made out of connections and is an "Art of variations" you will be rewarded. Those interested in the "frontier" between discrete and continuous will be interested too.
Definitely pedagogical and moreover intellectually honest.
... Read more

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When you need just the essentials of geometry, this Easy Outlines book is there to help

If you are looking for a quick nuts-and-bolts overview of geometry, it’s got to be Schaum's Easy Outline. This book is a pared-down, simplified, and tightly focused version of its Schaum’s Outline cousin, with an emphasis on clarity and conciseness.

Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give you quick pointers to the essentials.

• Perfect if you have missed class or need extra review
• Gives you expert help from teachers who are authorities in their fields
• So small and light that it fits in your backpack!

Topics include: Lines, Angles, and Triangles, Deductive Reasoning, Triangles, Parallel Lines, Distances, and Angle Sums, Trapezoids and Parallelograms, Circles, Similarity, Areas, Regular Polygons and the Circle, Constructions, Formulas for Reference, Proofs of Important Theorems

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This elegant little book discusses a famous problem that helped to define the field now known as graph theory: What is the minimum number of colors required to print a map such that no two adjoining countries have the same color, no matter how convoluted their boundaries are. Many famous mathematicians have worked on the problem, but the proof eluded formulation until the 1970s, when it was finally cracked with a brute-force approach using a computer. The Four-Color Theorem begins by discussing the history of the problem up to the new approach given in the 1990s (by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas). The book then goes into the mathematics, with a detailed discussion of how to convert the originally topological problem into a combinatorial one that is both elementary enough that anyone with a basic knowledge of geometry can follow it and also rigorous enough that a mathematician can read it with satisfaction. The authors discuss the mathematics and point to the philosophical debate that ensued when the proof was announced: Just what is a mathematical proof, if it takes a computer to provide one -- and is such a thing a proof at all? ... Read more