5 comments about Computational Geometry: An Introduction (Monographs in Computer Science).

1. I have just happened to exhume this book from my library, after it spent some years gathering dust above the shelf. In spite of the long time I have not being reading it, it still retains the full meaning it showed me when I was using in calculations relating radar domain definition. May be the textbook wins by far the comparison to the current vague and inflated computer publications, may be it is not a manager-oriented issue but it is for nearly specialistic use, you find in it clearly stated, and straight, answers to the questions you meet, or at least a definite reference where a more detailed explanation can be find. It presents interesting problems, and explains you how to solve them. I think it is the best you can say about a computer science book.



2. Most of the papers that I've read on computational geometry refer to this text -- and for good reason. There's many good algorithms to be found here.

   The book only gets 4 stars because it's hard to read. It took me several tries to pick up the ideas in this text. I think the De Berg text is MUCH easier to read.

   The book is also getting a little dated. Some of the topics have come a long way since the 80's.

   This book seems to be in most University libraries if you have that option.




3. The ideas and algorithms presented in this book are clear enough for straight implementation in code. I have long experience in developing comercial and production software for VLSI layout applications, which made extensive use of the algorithms presented in this book.
   I also use some chapters of this book as a part of a graduate course in VLSI layout algorithms being tought at the Technion, Israel. The contents of this book is well understood by EE and CS students.
   I personally love this book, which introduced me into the area of computational geometry and its applications.



4. This book is a classic, in fact the author's PhD thesis created this field, but this book is too old for any meaningful graduate work. There are new bounds and algorithms on almost all topics, which makes this a somewhat undesirable book. Also, this book has failed to keep me interested in it, while I am reading it...



5. This book is to computational geometrists what the King James Version of the Bible is to christian fundimenalists. Even though newer translations of the Bible are easier to read, somehow nothing sounds quite so authentically like the voice of God than those Elisibethen cadences, written in an almost archaic language....
   
   ...similarly for this book. Many times, the descriptions of algorithms presented in this book are made unnecesarily hard by very arcane langauge.
   
   But this book is authoritative and definitive in a way that no other text on computational geometry is ever likely to achieve. Even though there are any number of books which are newer and easier to read, it seems like this the one book on the shelf of every serious computational geometer I know.

5 comments about Topology (Undergraduate Texts in Mathematics).

1. Excelent, clear, well-motivated introduction



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

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

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

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




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

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

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




4. This text gives the reason behind many advanced topological concepts and tantalizes the reader with it's varied applications.
   
   Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.
   
   The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).
   
   Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.
   
   I especially enjoyed the section on covering spaces with which it concludes.
   
   Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.
   
   The text is useful to all students of mathematics and physics alike.



5. While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathemaicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the auther even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.
   
   Certainly this book couldn't be used as a textbook for an undergraduate course - for the reasons mentioned above and also because not enough material is actually covered, as well as the obvious deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated. The book also covers both point-set topology (topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.) and elements of algebraic topology (homotopy, fundamental group, simplices, CW-complexes, covering spaces, but not really homology), but the presentation of the algebraic topology in particular is not liable to be helpful for the novice, except for the treatment of covering spaces, which is perhaps the highlight of the book. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index:
   "We talk about homology (and a number of other objects beyond the realm of point-set topology) several times in this book, but the definition is not given."
   That, in a nutshell, explains the difficulty with this book.
   
   So why am I rating this 5 stars? For the wealth of examples (e.g., 4 sections on examples of quotient spaces) and explanations of how these concepts are used and why they are important. Just by looking at the contents one can see this, as there are sections entitled:
   "What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?," "The role of the countability axioms," "Why CW-complexes are more flexible," "Yes, but...?," "The role of covering spaces in mathematics," "What is it [Tychonoff's theorem] good for?"
   The chapter on covering spaces, coming near the end of the book, is particularly good, with a proof of their classification given. This is definitely the most fleshed-out part; if only the rest of the book could go into this depth.
   
   This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a subsitutute for it. But I would still consider this as a must-read for all those students who plan on studying mathematics in graduate school.
   
   Incidentally, a softcover international edition of this book is also available in India. The quality of the printing is not as good as the usual version, but the contents are the same, and it only costs about $11 plus shipping if you order from a seller in India. Generally international edition books from India are of a liitle better print quality than those from China.

5 comments about Mathematics and the Imagination.

1. Originally published in 1940, the material in this book is beginning to show a little age. However, the quality of the writing renders those defects to near irrelevancy. Popular descriptions of mathematics are differentiated by the quality of the writing rather than the distinctiveness of the mathematics, and this one shines.
   I like this book, starting with the title. It takes an enormous amount of imagination to do mathematics, something unappreciated by the public. It is easy to understand the use of linear segments to approximate the length of a curve. However, it requires an enormous leap of abstraction to believe that if they are made of zero length and then summed up, the result is the true length. Calculus students dutifully record and apply this, but in most cases don't appreciate the significance of the idea. In nearly all cases of major mathematical advancement, a fundamental change in thought processes was necessary. Those changes require imagination and the advances explained in this book are well documented and described.
   Mathematicians are containers of some of the greatest concentrations of imagination that humans possess. Their leaps of abstraction often include descriptions of objects that cannot be visualized. Kasner and Newman capture this essential ingredient, serving it up in palatable portions.



2. My school teacher gave me this book to read when I was 13 years old, based on the interest I showed in Mathematics that went beyond the curriculum at school. In many ways it was way beyond my comprehension at the time, but little did I know that it would have such a lasting effect on me. Reading about concepts of infinity, that you could only describe to a fellow teenager as "different sizes of infinity", I realized that there really is a philosophy of mathematics that transcends all other subjects and that there is also an art to working with the subject. I can't recommend this book enough, and I never did give it back to my teacher!



3. My only complaint is its lack of rigor and the fact that it is getting rather out-of-date; besides that, this is the sort of book that everyone interested in math should read while they're in high school.



4. If you like mathematics and how numbers and formulas work, it's worth having a look.



5. I am thoroughly enjoying the challenges to my brain that I am finding in this book. it was well worth the price I paid

No comments about Geometry, Study Guide and Intervention Workbook.

1 comments about Moduli of Curves (Graduate Texts in Mathematics).

1. The canonical strategy of modern mathematics when studying an object is to put this object into a collection, and see what properties they have in common. Most commonly, the objects depend on some parameter(s), and the goal is to find out how the objects vary with these parameters. The authors of this book take this approach to studying algebraic curves, with the parametrization being called the moduli space, and it enables one to gain information about the geometry of a family of objects from the moduli space and vice versa. The objects are typically schemes, sheaves, or morphisms parametrized by a scheme called the base. Putting an equivalence relation on the families gives a functor, called the moduli functor, which acts on the category of schemes to the category of sets. The functor is representable in the category of schemes if there is an isomorphism between the functor and the functor of points of a scheme. This particular scheme is called the fine moduli space for the functor, as distinguished from the coarse moduli space, where the functor is not representable, i.e. only a natural transformation, and not an isomorphism exists.

   The authors clarify the distinction between a moduli space and a parameter space, the former used for problems that involve intrinsic data, the latter for problems involving extrinsic data. An example of the latter, the Hilbert scheme, is discussed in detail in the first chapter, and an example due to Mumford of a component of a Hilbert scheme of space curves that is everywhere nonreduced is given to illustrate the pathologies that can arise in the extrinsic case, and to motivate the use of intrinsic moduli spaces to eliminate these difficulties. Severi varieties are discussed as objects that are more well-behaved than Hilbert schemes but still do not permit a scheme structure to be defined on them so that they represent the functor of families of plane curves with the correct geometric properties.

   The second chapter gives a general overview of the approaches taken in the construction of moduli spaces of curves. The authors first study the case of genus 1 (elliptic) curves to illustrate the difficulties involved in constructing fine moduli spaces. The role of automorphisms on the curves as an obstruction to the existence of fine moduli spaces is outlined, as well as approaches to deal with these automorphims, particularly the role of marked points. The authors briefly discuss the role of algebraic spaces and algebraic stacks in the moduli problem. They explain also the various approaches to the construction of the moduli space of smooth curves of genus g, namely the Teichmuller, Hodge-theoretic, and geometric invariant theoretic approaches. The local properties of the moduli space are outlined, along with a discussion of to what extent the moduli space deviates from being a projective or affine variety. The rational cohomology ring of the moduli space is also treated, in low dimensions via the Harer stability theorem, and for high dimensions via the Mumford conjecture. Most interestingly, Witten's conjectures and the Kontsevich formulas are introduced, as a theory of moduli spaces of stable maps. The famous Gromov-Witten invariants of a projective scheme and the quantum cohomology ring are briefly discussed. These have generated an enormous amount of research, the results of which show the power of viewing mathematical constructions from a "quantum" point of view.

   The next chapter gives a very specialized overview of the techniques used to study moduli spaces. The authors are very meticulous in their explanations of where the names of the concepts come from, and this is an enormous help to those seeking an in-depth understanding of the topics. One of the first is the dualizing sheaf of a nodal curve, which is the analogue of the canonical bundle of a smooth curve. The authors then describe, by taking a point as the base, the scheme-theoretic automorphism group of a stable curve, and show that it is finite and reduced. Deformation theory is introduced first as over smooth varieties. Readers will appreciate the discussion more if they have a background in the deformation theory of compact, complex manifolds. The authors then tackle the stable reduction problem, and give several beautiful examples, with lots of diagrams, to illustrate the concepts. This is one of the best discussions I have seen in print on these topics. After a brief interlude on the properties of the moduli stack, the authors treat the generalization of the Riemann-Roch formula due to Grothendieck. This section is very important to physicists working in superstring theory. The Porteous formula is also stated and applied to the determination of the class in the rational Picard group of hyperelliptic curves. The determination of the class of the locus of hyperelliptic stable curves of genus 3 is continued in two more sections using the method of test curves and admissible covers.

   The actual construction of the moduli space is the subject of chapter 4, from the viewpoint of geometric invariant theory. A nice example of this approach is given for the case of the set of smooth curves of genus 1. The numerical criterion for stability is discussed in detail, with Giesecker's criterion given the main focus. The case of the moduli space of curves with genus greater than two is tackled via the potential stability theorem.

   The authors show indeed in the next chapter that the moduli space can be used to prove results about a single curve. As one would expect intuitively, the taking of limits must be justified, and indeed this is the case here, where limits of line bundles and linear series are considered.
   Then in the last chapter they show the reverse, that the properties of various moduli spaces can be proven using the techniques introduced in the book, such as the irreducibility of the moduli space, the Diaz result that complete subvarieties of the moduli space have dimension at most genus - 2, and moduli of hyperelliptic curves and Severi varieties.

2 comments about Basic Algebraic Geometry 1: Varieties in Projective Space.

1. This book is very good for the secondary course after learning with Harshorne's Algebraic geometry.



2. I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start. Having said this, one must have the necessary background in algebra and topology. I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!! Oh ya, the index refers to both volumes 1 and 2; read the first page of the index!!!

4 comments about Differential Geometric Structures (Dover Books on Mathematics).

1. I have been recommending this book to my colleagues and students since 1981. Finally, they can get a copy easily.
   
   Prequisites are modest, and should be part of the standard math graduate curriculum anyway: the equivalent of Chapters 1 --3 of Warner (differential manifolds, tensors and forms, and a minimal introduction to Lie groups).
   
   Given these, it is simply the best introduction ever written.



2. I learned a great deal from this book
   in my second year of grad school and
   have recommended it to dozens of people
   since then. It is wonderful to see it
   back in print. A fantastic introduction
   to differential geometry.



3. this book is not for engineers, there is no introduction to those
   topics mentioned, if you do not have some mathematical background on manifolds etc, the book will not help you. i m giving 4 stars
   just to warn the engineers like me trying to get into differential geometry.



4. This book contains material about differential geometry that is very hard to find in any other book, if possible at all. However for people who feel uncomfortable with different approaches of what they already know, a word of warning is in order: the book builds everything on 5 axioms about parallel structures in bundles. All other approaches (frame bundles, connections e.t.c.) are deduced later in the book. Five axioms might take a hard-swallowing and the trade-off is that they mimic the intuition for Euclidean spaces. Of course any serious reader will not expect to learn differential geometry from one book, so overall it is a useful addition to your collection. Finally, looking at the size of the book and the material it covers you can expect the text to be pretty dense and this is actually the case. Several books on smooth manifolds are suggested in the beginning as companions and Warner's book, which is cited consistently for background results, is a prerequisite for this book.

3 comments about Catastrophe Theory and Its Applications.

1. This book gives many informations of applications of Topology. With their illustrations give us the more intuition and help us for more understanding with a situation in the reality life. AT First half of book, we can remember the theories of Calculus, Transversality and Stability and then brings us to how we can see the application these theories in our life, examples, in the domain of biology, physics,ecology and social. I think it's a good book for a student who works not only,in Differential Topology , for seing their applications and give an idea how we can apply the theory what we studied,but for the post graduates students in enginneering,physics and biology also.



2. easy understand theory and applications



3. If you want to get started into catastrophe theory, this book is an excellent place to start. It gives you a quick idea into some of the key concepts with less investment of time. But if you are ambitious, you want to look elsewhere like some of Arnold's books.

2 comments about Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Mathematical World).

1. Groups are the first structures encountered in abstract algebra and form the foundation for most of the others. Fortunately, they are also the easiest to physically represent, so in some sense they are the most concrete. In this book, groups are introduced as the motions and structures of geometric figures, so the presentation is largely by diagram rather than formula. Very little previous knowledge of mathematics is required and after reading the book, you will have a solid understanding of what a group is.
   The first topic is the moving of a complete figure to a different location of the plane defined by a grid of points. By keeping the figure rigid and fixed in orientation, a set of legal moves is defined. After that, some of the rules are relaxed and that allows for additional moves to be added. Exercises and problems are put forward here and throughout the book, and with the accent on figures, often give the appearance of a game.
   The next steps are then to allow for all possible rotations, translations and reflections of the objects, using these to explain the structure of a group. This is an effective way to introduce group theory, and is how I will do it if I teach abstract algebra again. Permutation and plane tiling symmetry groups are then introduced and examined, and their relationship to the previous groups discussed, which introduces the concept of isomorphism.
   Basic group theory is something that everyone can understand, as humans have a natural affinity for patterns and recognizing them despite "trivial" alterations. This book is an excellent primer on group theory and I strongly recommend it to anyone either learning or teaching abstract algebra.

   Published in Journal of Recreational Mathematics, reprinted with permission.




2. This book was the foundational textbook for a 100-level class in symmetry at my university. I recommend it highly to anyone who wants to get a better feel for what mathematicians actually do and think about and work with. Folks who never got into the higher math classes often have a different idea of what mathematics is all about than mathematicians. At the level of introductory algebra and geometry and even some calculus, math education often seems to be mainly about memorizing formulas and recognizing in which situations to apply them. That's an important thing to learn, but it is not useful for imparting an idea and a feel of the field of mathematics as a whole. Farmer's book brings home the understanding that mathematics is, at its heart, about patterns and that mathematics is not so much about memorization and application as it is about discovery.
   
   The level of mathematical understanding required to get something useful out of this book is low. I believe the professor required beginning algebra as the prerequisite. If you can count to six, recognize the difference between a square and a pentagon, and understand that variables like n, m, or x can be used as substitutes for numbers then you probably have enough mathematical sophistication to work your way through this book and gain insights into the beauty of higher math.

1 comments about Elementary Mathematics from an Advanced Standpoint: Geometry (Dover Books on Mathematics).

1. About the set: Klein's Elementarmathematik lectures was intended as a survey of mathematics for those who already knew most of the technical detail, especially future teachers, but who perhaps lacked a good understanding of mathematics as a whole. The lack of a broad perspective is probably at least as big a problem today as it was then, so Klein's text is still valuable. Klein also frequently discusses historical and pedagogical aspects, and the tone is quite informal throughout.
   
   About this volume: The first part is an elementary introduction to vector techniques in geometry, which is probably not very interesting for the modern reader. The second part of the book is more interesting. Klein has explained in his preface that he wants to present a unified view of geometry, and here in the second part he shows how many diverse concepts can be brought together under the notion of transformation. This approach paves the way for a discussion of Klein's own Erlanger Programm in part three. This third part is very nice. First there is a "systematic discussion of geometry", where the Erlanger Programm is one of the themes, then there is a discussion of the foundations of geometry. This part of the book makes great pleasure reading, as Klein speaks freely and informally, and sometimes voices his personal opinions.