1 comments about Elementary Algebraic Geometry (Student Mathematical Library, Vol. 20) (Student Mathematical Library, V. 20).

1. First, a calibration: I am a total neophyte to algebraic geometry, and haven't taken a university algebra course since a few decades ago when I was a physics major. This book is one of several on the subject (along with some books on commutative algebra) that I'm using to get an amateur's orientation.
   
   As so often happens, this book looked great in the bookstore. It is thin, reasonably well-illustrated compared to other books in the field, and even helps you gets your toes wet in sheaves, category theory and some other neat topics.
   
   That said, I believe the prerequisites in the preface (university algebra, with a complex variables course optional) are understated; e.g. it helps to know something about fibres, lifts and other topics from geometry. It might be relevant that these notes were prepared at a German university; you should consider that "undergraduates" there are heading toward the equivalent of a US M.S. degree, not B.S./B.A.
   
   More detrimental is that the presentation slogs from one proof to another and too rarely pauses for breath to consider the "big picture" significance of what you're proving. Notwithstanding that Joe Harris's "Algebraic Geometry: A First Course" is even less of a piece of cake for me than it might be for you, his style is a breath of fresh air when it comes to enlightening you as to some geometric context and payoff for all this effort. Other supplements I found helpful include Reid and Schenck.
   
   PS in 2008: I very belatedly found the terrific "An Invitation to Algebraic Geometry," by Karen E. Smith &al. (Springer 2000, corrected printing 2004). This is the hands-down best introduction to the subject, IMHO.

4 comments about Symmetry in Mechanics.

1. There are a number of books available on the "geometric" view of physics (Classical Mathematical Physics, by Thirring, The Geometry of Physics, by Frankel, and Foundations of Mechanics, by Abraham & Marsden). The size, level of sophistication and extensive background assumed by these books can be very intimidating. On the other hand, the subject "looks" beautiful, and the benefits of using geometric intuition are desirable to many people.

   Singer's book stands class of its own in these respects. All the basics of the geometrical "machinery" are there, in a book that is only 224 pages in length. Chapter one starts with a standard derivation of the equations of the "two-body planetary motion" problem; subsequent chapters proceed to introduce the necessary modern geometrical and mathematical concepts (differential geometry). The final chapter then revisits the "planetary motion" problem using the modern concepts previously introduced. Excellent!

   There are some misprints, but the author has a Web page of errata. The book has numerous exercises, with many solutions included. I find myself rereading parts of this book over and over.

   Reader be warned; the concepts are new, and it does take work to internalize them. However, this is the most accessible book on the subject available, and also one of the most affordable. The author references many other books, for the reader who wants to go further in the mastery of this subject (one excellent book which is not mentioned, however, is "Differential Forms: A Complement to Vector Calculus", by Weintraub). Enjoy!




2. This is one of the very few books which I returned for refund.
   The subject is intrinsicly interesting, and there is a need for a serious introductory text addressing the subject of geometry and physics. This one badly falls short, - carelessly written, with numerous irrelevant asides. She seems even to fail to realize that there exist three distinct geometric solutions to the Kepler problem. The bound, elliptic case is only one.
   This book has supposedly been written for high-undergraduate students or early-year graduate students. It serves neither adequately.



3. I think the previous review is a bit harsh, and that the book's intents are not what this reviewer expected. I don't think it was the author's intent to write a comprehensive treatise on the subject. The book simply aims at introducing undergraduate students to the use of symmetry in simplifying the analysis of classical mechanic problems, nothing more. If you want a comprehensive treatise, you probably want to read V.I. Arnols's "Mathematical methods in classical mechanics". If what you want is a simple introduction where all the steps are worked out in details, then this book is a good starting point, and I think this is what the author intended. At any rate, the cost ($$$) is quite reasonable.



4. There are two classes of books in mechanics: the extremely physical, which are intended to teach you how to solve problems but lack any mathematical rigour, and the mathematical ones, where the examples are generally one-line statements without any explanation. This book sits exactly in the middle of both: if you are a physicist (or mathematician for that matter) with a fair knowledge of classical mechanics and you understand the basics of Hamiltonian systems, but you want to expand your horizon with momentum maps and symplectic reduction, but you don't understand anything of the hardcore abstract books by mathematicians or you are afraid of them, this is where you should put your money.
   
   Physicists usually simplify their equations by using symmetry in a rather ad hoc way; intuition tells you that a rotation around a certain axis does not change anything or that the system is invariant under translations, or that angular momentum is conserved in a certain direction. Symplectic reduction is the systematic study of these symmetries and how to simplify you equations with them. Don't expect to be shocked because most of the analyses can be carried out without knowing anything about symplectic reduction, but it can aid your life if you are working on more complicated systems, where your intuition does not help you very much (or if you just want to impress someone with your knowledge of mathematical mechanics).
   
   The book does not go deeply into the material, but it explains the basics clearly (symplectic two-form, momentum maps, Lie derivative, reduction...) without being pedantically mathematical. Don't expect any proofs or general theorems; e.g. the author uses (dual) MATRIX Lie groups/algebras, which are intuitive for the physicist (just apply the matrices to your coordinate basis and that's it, quick and dirty) but not as general as the idea of coadjoint orbits of an abstract Lie algebra.
   
   I have tried to go through the mathematics library on symplectic topology and symplectic reduction but have never come very far - and in the cases I thought I understood the concepts I found out that I could do absolutely nothing with it in practice, because I had never seen an actual calculation. After reading this book I must say that I have more confidence reading and understanding them. The book prepares you for more to come, which is exactly what it's aimed at. Instead of giving you the dry reality of modern mathematics wrapped in complete generality, it gives you the juicy extract of what it's all about, it lets you think about it, and use it in simple situations. If you want to go beyond this book, you'll have to have a firm knowledge of Lie groups, Lie algebras, and differential geometry, but for this book, you just need undergraduate physics and mathematics.
   
   The book comes with lots of exercises and to some the answers are given at the back. It's a short and easy introduction to the uses of symmetry (reduction) in Hamiltonian mechanics, and it's good value for your money. I am happy to have it and I can only recommend it.

No comments about Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors (Cambridge Texts in Applied Mathematics).

No comments about Girolamo Saccheri's Euclides Vindicatus.

No comments about Non-Euclidean Geometries: János Bolyai Memorial Volume (Mathematics and Its Applications).

1 comments about Methods of Information Geometry (Tanslations of Mathematical Monographs).

1. This book is a comprehensive introduction to Amari's approach to information geometry (IG).
   
   The book starts with an introduction to differential geometry (DG). This chapter of the book is difficult to follow for those who have no background in differential geometry. Besides, the introduction contains detail definitions of advanced concepts that could be skipped in the first run. So, as an engineer, I would suggest my fellows to consider another books, e.g. by "W. M. Boothby" or "M. Do Carmo" for building the necessary background.
   
   After the introduction, the statistical models and the fundamental notion of Fisher information and the most important features of Amari's IG, i.e. dual connections, are introduced in the second and third chapters. Reading these two chapters, needs a certain amount of patience for those engineers like myself who are more "goal-oriented".
   
   Finally, the forth chapter provides the application of previously explained methods to the statistical inference and estimation. This part of the book is very informative, although not so smooth to follow. Despite the fact that I had to go back and forth through out this part of the book to find a smooth and thorough understanding of the concepts, I really enjoyed reading this part. The last part of the chapter focuses on more advances topics like higher-order asymptotic statistics (which might be not that necessary for many of engineers who mainly like to talk about first and second order statistics), and fiber bundles which could be skipped for many of us.
   
   The last part of the book provides some examples for the application of IG, e.g. time-series analysis and identification of the linear systems, multiterminal distribution, IG of quantum information, convex optimization, ... These last four chapters could have been published as another editorial book with more details. This part of the book is useful for those who need some motivation for getting involved with IG. A comprehensive list of references is provided for serious readers who want to dive into the subjects.
   
   All in all, Amari's book on IG is a "must-have" book for those interested in information geometry and differential geometry of statistics. The book is my daily reference for my research, as I learn more of it everyday. The book is useful for a wide spectrum of readers, e.g. for the beginners to find the big picture of IG, for advanced readers to find Amari's taste of IG, and for engineers to learn the fundamentals, see some applications, and use the results in a reasonable way.
   
   It is important to mention that there are other approaches to IG. As an engineer, I have found this field very sophisticated and mature and sometimes confusing. So, before using the results of the book, it is a wise decision to always consider other references and consult the experts in the field.
   
   Amin Zia
   PhD Student
   McMaster University
   Hamilton, Ontario
   Canada

1 comments about An Introduction to Differential Manifolds.

1. This book is just so full of useful information and details. It has a lot of problems for which most of the solutions are supplied. Man, I love differential manifolds after spending some quality time with this book.

5 comments about The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Springer Monographs in Mathematics).

1. A new edition of this book is available from Springer now, for a lower list price (and one third off if you have ever published in a Springer publication, like LNCS, LNM etc.).



2. I'm a graduate student in set theory and I'm finding Kanamori an excellent follow-up to Kunen. The book manages to combine detailed technical exposition with historical insight which is actually useful to understanding the material (not just a list of dates) and gives one a "feel" for the subject.
   
   Occasional excersises are contained which are good to help check if you're keeping up (though sometimes the hints are a little too complete: it might be better if these were relegated to an appendix). More exercises would have improved this book.
   
   I believe this is pretty much the only book in which much of this material is collected together, so it's pretty much essential to any-one seriously interested in Set Theory. I await the promised second and third volumes with anticipation!



3. Assumes set theory through forcing and some model theory. He integrates a lot of historical information which is interesting but greatly adds to the reading without adding to the mathematical understanding. The semester is short and I want to learn what I need and get on with life. While it does have more details than Jech, I like Jech's style better and not just because it is shorter. (The 2e is just a corrected reprint of 1e.)



4. A must have for set theorists. Does a better job than Jech. It's a little weird how you get a sense of the set theorists as people, but the historical stuff is otherwise really great.



5. If you have background in set theory the book is ok if not then it's difficult reading

No comments about Renormalization and 3-Manifolds Which Fiber over the Circle.

No comments about Topology I: General Survey (Encyclopaedia of Mathematical Sciences).