Book Description

The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his co-workers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings' proof of the Mordell Conjecture).

This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required. ... Read more

Reviews (3)

Crystal clear overview of a traditionally abstract subject
The theory of schemes is usually thought to be highly abstract and esoteric, and one that makes the study of algebraic geometry even more difficult. The authors definitely dispel this notion in this book, which could have been called "A Concrete Introduction to Schemes", because of the clarity with which the concepts are introduced and explained. After studying this book, one will understand and appreciate the power of schemes in algebraic geometry. The authors do an even better job than they did in their earlier and short work "Schemes: The Language of Modern Algebraic Geometry", which is now out of print.

In chapter 1, the main definitions are given and the basic concepts behind schemes outlined. That schemes are more complicated than varieties is readily apparent even in this beginning chapter, where they are thought of as corresponding to the spectrum of a commutative ring with identity. Very elementary exercises are given to help the reader gain confidence in the constructions involved. They authors do have to discuss some sheaf theory, but they show its relevance nicely in this chapter. They also discuss the notion of a fibered product as a generalization of the idea of a preimage of a set under the application of a function and relate it to the construction of the functor of points. The role of the functor of points as reducing schemes to a kind of set theory is brought out beautifully here.

The next chapter gives many examples of schemes, with the first examples being reduced schemes over algebraically closed fields, these being essentially the ordinary varieties of classical algebraic geometry. The authors then give examples of schemes, the local schemes, which are more general than varieties. When departing from the assumption of a field that is not finitely generated, extra points will have to be added to classical varieties. The fact that only one closed point appears is compared to the case of complex manifolds, via the concept of a germ. This is a very helpful comparison, and one that further solidifies the understanding of a scheme in the mind of the reader. The authors give the reader a short peek at the etale topology in one of the examples. Examples are then given where the field is not algebraically closed, generalizing classical number theory, and non-reduced schemes, where nilpotents are present. The chapter ends with examples of arithmetic schemes where the spectra of rings are finitely generated over the integers.

Projective schemes are the subject of Chapter 3, and are defined in terms of graded algebras and invariants of projective schemes embedded in projective space are discussed. The Grasmannian scheme is discussed in detail as an example of a projective scheme. Interestingly, Bezout's theorem, very familiar from elementary algebraic geometry, is generalized here to projective schemes.

Constructions from classical algebraic geometry are generalized to schemes in Chapter 4. The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety. The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes. The notion of blowing up is also generalized to the scheme setting, with the authors motivating the discussion by blowing up the plane. The discussion of blow-ups along non-reduced subschemes of a scheme and blow-ups of arithmetic schemes is fascinating and the presentation is crystal clear. Fano varieties are also generalized to Fano schemes in the chapter. Most of the information about these schemes are contained in the exercises, and some of these need to be worked out for a thorough understanding.

The next chapter is more categorical in nature, and deals with generalizations of the classical Sylvester construction of resultants and discriminants to the scheme setting.

In the last chapter the authors return to the functor of points, and motivate the discussion by asking for a parametrization of families of schemes. The authors show, interestingly, that using the functor of points one can more easily compute geometric information about a scheme than using its equations. They illustrate this for the Zariski tangent space. Then after an overview of Hilbert schemes they close the book by introducing the reader to moduli spaces and a hint of algebraic stacks. No end in sight for this beautiful subject..........

A very good start
This book is clear, well written, and has a nice balance of generalities and examples. If you know the basics of rings and modules, this book will show you what schemes are and why they are useful for several different problems: for example, number theory, or studying singularities. I find it a helpful companion to Hartshorne's ALGEBRAIC GEOMETRY. But this book does not get to cohomology, and so cannot actually get to the working methods in the subject. For that, you need Hartshorne.

Very good book
Very good book for scheme theoritical approach to Algebraic Geometry ... Read more

Book Description

Combinatorial commutative algebra is an active area of research with thriving connections to other fields of pure and applied mathematics. This book provides a self-contained introduction to the subject, with an emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determinantal rings. The eighteen chapters cover a broad spectrum of topics, ranging from homological invariants of monomial ideals and their polyhedral resolutions, to hands-on tools for studying algebraic varieties with group actions, such as toric varieties, flag varieties, quiver loci, and Hilbert schemes. Over 100 figures, 250 exercises, and pointers to the literature make this book appealing to both graduate students and researchers.

Book Description

Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. ... Read more

Reviews (5)

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Good attempt
When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the concepts of differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

Covers a lot of ground . . . but not always well
Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists. I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered. However, if a proof actually is presented, I expect this proof to be correct. In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

flawed and incomplete
Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else.

Great introduction to mathematical physics.
This book is written by physicists. Like a book by M. Nakahara it describes basics of diff geometry and topology. Though it stresses physical intuition more than formal definitions. I especially liked discussion of fiber bundles and characteristic classes. Highly recommended. ... Read more

Book Description

The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature.

Key features:

* Basic results presented in an accessible way for both pure and applied mathematicians

* Extensive exercises make the work suitable as a textbook for use in graduate courses

* Full proofs included in introductory chapters; only basic knowledge of functional analysis required

* Explicit constructions of frames with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series

* Selected research topics presented with recommendations for more advanced topics and further reading

* Open problems to stimulate further research

"An Introduction to Frames and Riesz Bases" will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference. ... Read more

Reviews (1)

A new tool!
The subject of the book is a new tool in math, with a host of exciting applications. Of course, the subject has roots in classical ideas from harmonic analysis. But the book covers an explosive and exciting variety of developments since roughly 1990, and it is presented in the form of a graduate text. The basic idea begins with linear algebra, and progresses to expansions in function spaces, and multiresolutions. It will be useful to anyone who wants to learn from scratch about the underlying principles, the new results, and the applications. It is well written. A student of mine picked it up accidentally from my desk, and couldn't put it down. After awhile, he had completely forgotten what he came to see me about. It could have been some of the applications, such as antenna theory, wavelets, time-frequency analysis, uses in Radar, speech processing... ... Read more

Book Description

A fresh approach to introductory topology, this volume explains nontrivial applications of metric space topology to analysis, clearly establishing their relationship. Also, topics from elementary algebraic topology focus on concrete results with minimal algebraic formalism. The first two chapters consider metric space and point-set topology; the second two, algebraic topological material. 1983 ed. Solutions to Selected Exercises. List of Notations. Index. 51 illus.
... Read more

Reviews (4)

excellent introduction to topology
I used this book to teach myself the basics of point-set topology and homotopy theory. What makes this book so great is that the author doesn't waste words in delving into the heart of a concept, while providing insight into it. A good collection of interesting problems, most with solutions in the back of the book. This makes this book very good for self study. If you liked Rudin, you'll probably like this book as well, as it is written in a similar style. If someone knows of a better introduction, do let me know.

exceptionally well organized
This is a lean fast introduction to topology at the third or fourth year level. Pure math types only. The book is terse but the topics are selected with care and one things leads to the next. The proofs are sufficiently detailed. Nearly every exercise has a solution in the back. The clearest exposition of the fundamental group I've seen.

Good grad school prep.
This is the usual text for introductory Topology at UCLA, where I took the course. The authors (who teach at UCLA) have "if you haven't chewed through every syllable you are not learning" mentality. In short, the book is terse and demands a lot from the reader. Looking back, this was great preparation for graduate school and is probably the best philosophy for the serious undergrad. The book contains all of the information one needs for an introductory course, but absolutely no more. Not a single character is wasted on "extraneous" explanation. Be ready for battle when opening this one, but it's worth it.

A solid introduction
This book is divided into two parts: an introduction to topology, using metric spaces to motivate the definition of a topological space, and the algebraic applications of topology (such as homotopy theory and Jordan's theorem). The good things about it is its strong geometrical flavor and short and to-the-point explanations, which are nevertheless very good. Another excellent thing is the large number of exercises, most of which have full solutions (!) and/or hints at the end of the book. The chapter on metric spaces is also very informative, and stresses the many uses these spaces have in mathematics. However, the book does not really go into detail, especially when dealing with topological spaces. This is fine if you're looking for an introduction to topology (as the name implies . . . ) but it makes the book unsuitable as a reference. All-in-all a very "cute" book - as cute as introduction-to-topology books get! ... Read more

Book Description

This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems. ... Read more

Reviews (1)

Nice introduction to differential topology
This book introduces the basic concepts in differential topology, a field that has taken on particular importance in medical imaging, game theory, and network optimization. Although written for mathematicians, and therefore somewhat formal, a good course in multivariable calculus should prepare the reader for this book. The most difficult chapter is probably Chapter 2, where Hirsch studies manifolds by means of function spaces and jets. He does do a good job in this chapter though of explaining the origin and need for partitions of unity and gives examples. He also gives the reader good insight into why analytic maps are more difficult to handle than the C-r case, and wets the readers appetite for further reading on the analytic case. The important notion of transversality is discussed in Chapter 3, which would be good reading for one interested in applications of differential topology to dynamical systems. A more detailed discussion of vector bundles would have been helpful in Chapter 4, which discusses these important objects and the idea of a tubular neighborhood. Sring theorists or those learning the mathematics should get a lot out of Chapter 5, wherein intersection theory in differential topology is discussed. The most important chapter of the book is Chapter 6, which discusses Morse theory. The applications of Morse theory are immense, and cover not only mathematics, but physics via quantum field theory and string theory, economics, and even computer graphics. A short chapter on cobordism follows, which is very nicely written, but a few more words would have been nice on this topic. After discussing isotopy in Chapter 8, Hirsch gives a good proof of the classification for surfaces in the last chapter of the book. A nice book to have for reference if one is interested in the subject for its own sake or for its many applications. It should prepare one for further advanced reading in differential topology, such as the work of Freedman and Smale on the Poincare conjecture in dimesions 4 and above. Those interested in applications of differential topology will be amply prepared to apply these results to the relevant areas, which are many. ... Read more

Book Description

The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.

Line and surface integrals
Divergence and curl of vector fields ... Read more

Reviews (2)

great introductory text
My first course on manifolds was based on this book, and I believe that it is the best introduction to the subject (especially for beginners). I thoroughly enjoyed it! I should also recommend Conlon's 'Differentiable Manifolds' (2ed, Birkhauser), as it is the perfect follow-up to Boothby. --A

Very Nice Nontrivial Introduction
This book is a careful treatment of the subjects in the title. It is an introduction, but it manages to cover quite a bit of ground with lots of examples to illustrate. One of it's distinguishing points is the way in which the concrete, coordinate based calculations are emphasized even while usually presenting the more abstract, coordinate free approach as well.

The book does a good job at stimulating those studying it to develop intuition. I found the book helpful when I was first studying the subject. ... Read more

Book Description

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions. ... Read more

Reviews (6)

Integrability conditions discussed
Written in a attractive and even seductive way, relying more on Lie algebraic language than is typical, this book is probably as stimulating an intro. to modern geometry as you can find, within certain limits. The section on noncoordinate bases might have been more clearly written, however. Frobenius's theorm is discussed, something that Fomenko et al should have covered, and the section on connections can be worked throuigh independently of the heavy machinery of exterior differential forms, which is attractive for physics students.

Not as good as "a first course in general relativity"
I had read first the "first course in general relativity"and was exited,so i fygured out that this book from the same author would reach the same standards,but it didnt.If Ihadnt read the first book from Schutz this book would be incomprenheceble.The greatest problem i think is the lack of exercices.Without them you cant really go anywhere.Another problem ,i believe,is the short space given to analyzeeach topic.Eventhough i understand tensor calculus very well I just cant get anywhere with the differential forms.
Eventhough its not the worst book out there its not the best either.My advise,buy a better book.

A Very Accessible Book ! Buy It !
This is a very enjoyable and clearly written book. From a physics point of view the approach is rather abstract, so although differential geometry is developed from 'scratch', it is probably better to have studied a more elementary text on the theory of 2-surfaces in 3-space first (eg Faber's book Differential Geometry and Relativity Theory ). The first chapter sets the mathematical background expected of the reader. The rudiments of analysis, topology, calculus of many variables and basic linear algebra is reviewed.The ensuing chapters cover differential geometry from a 'modern' viewpoint but the style is quite relaxed and the links to 'co-ordinate approach' are well explained. The exercises concentrate on the abstract approach. Throughout the book the underlying structure of manifolds is concentrated upon. No extra 'structure' eg connections and 'distance' concepts are added until the final chapter on Riemannian spaces. For example the metric tensor throughout the body of the book is merely used as a map between a tangent space and its dual space. It is only used as a 'distance' operator in the final chapter.For the purposes of independent study this is a sound book, there are hints and partial solutions for many of the exercises, which is always a welcome feature for those studying entirely on their own.

A Great Introduction to Diff. Geometry
This book presents the basic concepts of differential geometry in a clear, concise manner using modern notation. Schutz's writing style is very readable and there is a considerable breadth of coverage. In areas where one might wish for greater depth, Schutz provides excellent references. My only regret is that the physical applications chapters weren't longer. An excellent starter book and a good quick reference if you continue in differential geometry, GR or field theory.

a little time and no love
I am very angry with paying this much money for a book which obviously dosen't live up to it's name. Think I that Schuz's writing "exposition" is no more than symbolic buffoonery. If you want a book that will knack ya sax off than get a copy of Schutz's "A First Course in General Relativity". ... Read more

Book Description

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientsts, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continum physics.This compendium contains material that was previously available only in specialist journals.This is likely to become the standard reference work for all those interested in the subject. ... Read more

Reviews (1)

An easy call
Very simply: if you want to know a very great deal about topos theory, buy this book. I mean, seriously, if you plan to make real work on topos theory a part of your life, then grit your teeth and come up with the money. If you do not want to know a very great deal about it, do not buy this book. You can use it at the library as a reference.

If you merely want a professional understanding of what topos theory is, then read Johnstone's earlier TOPOS THEORY. That far shorter book gives a better overview. My Amazon review of it discusses others on the subject. Most are more accessible than Johnstone's books and go more into particular aspects of the theory.

This book is a reference on all the methods, and the latest results, in topos theory. If you want the definition of "split opfibration", it is here, along with some 80 pages of background, examples, and motivation. Johnstone does an heroic job of unifying the terminology and organizing the theorems.

More than that, Johnstone has written down an expert, encyclopedic view of the subject today. It is rare for a top mathematical researcher to give so deep an account of their field. It is rare for anyone to even work out such an explicit, coherent, extensive account of the whole. Not everyone will agree with his view. Some would like to see much less of such logical topics as "allegories", others would like to see the logic more formalized from the start. But Johnstone builds a case for his choices: partly implicit in his success at explaining things this way, and partly by explicit reasons.

If you want to know that much about the subject then you want to immerse yourself in this book. ... Read more

Book Description

This book describes many new results and extensions of the theory of generalized topological degree for densely defined A-proper operators and presents important applications, particularly to boundary value problems of nonlinear ordinary and partial differential equations that are intractable under any other existing theory. A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation.The theory subsumes classical theory involving compact vector fields as well as the more recent theories of condensing vector-fields, strongly monotone, and strongly accretive maps. Researchers and graduate students in mathematics, applied mathematics, and physics who make use of nonlinear analysis will find this an important resource for new techniques. ... Read more

Book Description

Cinderella is a unique, technically very sophisticated teachware for geometry. It will be used as a tool by students learning Euclidean, projective, spherical and hyperbolic geometry, as well as in geometric research by scientists. Moreover, it can also serve as an authors' tool to design web pages with interactive constructions or even complete geometry exercises. For this Version 1.2 a few minor items have been changed, such as: - you can add greek letters in text - the dialog window for exercises has changed - text objects work in hyperbolic and spherical ports, too - the MacOS installation procedure has changed. ... Read more

Reviews (6)

Not a rip-off; please contact Springer Support.
Hi,

this is just an answer to the complaint found below. I am sorry for the current state of our website, but you should not mix-up our website with the support you get when you buy Cinderella. The web site is done in our spare time; the support is handled by Springer-Verlag. In your case, just contact Springer at service@springer-ny.com -- they can help you with the Windows XP issues you are experiencing.

Ulli Kortenkamp (one of the authors of Cinderella)

Nice Box, No Support
The box is very enticing, but the software will not run on Windows XP or Windows 2000. The "support" turns out to be a web site that is under construction and is full of broken links. The publisher (Springer) does not seem to offer support either, they refer you to the same useless web site.
Some of the provided precompiled java examples were interesting, but I never could get the package itself to run.
Without proper support, this product is a rip-off and should be withdrawn.

Good concept, but poorly supported
I was impressed by this software when I first bought it.The graphics are nice, the interface is clean and intuitive.But there are many features lacking: the ability to put arbitrary text in the construction, to draw arcs of circles, to make calculations with measurements and much more.Several of these features are promised `in the next update' on the product's website, but this update has never appeared, nor is there any timetable for expected upgrades.Another problem I have had with this software is in installation: whether installing to Linux, recent Windows or OS X, there are always difficulties -- again (I suspect) because the software writers have not been responsive to the problems with some java platforms.

There are many free, open source alternatives to this product: for instance, the graphical KGeo (for Unixen),C.a.R. -- Compass and Ruler (java), and the text-based eukleides (Unixen and Windows).These have all the features of Cinderella and more; and, most importantly, they are actively supported: bugs and feature requests are dealt with.

Cinderella interactive geometry review
Cinderella is very similar to Geometer's Sketchpad.There are some important differences.Cinderella uses more sophisticated algorithms to draw objects and therefore, some minor annoyances of Sketchpad are eliminated.In sketchpad, objects defined as the intersection of two circles may jump from one intersection to another under some conditions.This does not happen in Cinderella.The only shortcoming of Cinderella (in comparison to Sketchpad) is that Cinderella does not do computations.That is, you cannot display the product of two segment lengths.Cinderella works very smoothly, and is half the price of Sketchpad.I like them both, but would go for Cinderella based on price.

Great Geometry on a Computer
I have played around with other software for making geometric constructions, but this is by far the easiest to learn.The basic options are all listed for you at the top of the screen, so that they are easy toaccess and with a little practice very complex geometric designs can beeasily created.Once your design has been created you can grab a point,move it around, and see the effect on other points in the design.The"locus" feature is fantastic for understanding how the motion ofone point effects other points in the figure.Your control over point andline, color and size is outstanding. The line clipping function and theability to make parts disappear while still playing their function makes itpossible for you to create designs which demonstrate your ideas in a clearmanner.The printed results, with respect to lines and circles, is for themost part outstanding.I congratulate the authors.The price is right. Buy it. ... Read more

Book Description

Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups, or probability or differential geometry. It is ideal both as a reference and for those wishing to enter the field. ... Read more

Book Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1 ... Read more

Reviews (13)

over and over and over again
Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day.

Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain.

For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.

the geometry of physics
I just finished a class in mathematical physics, and the text we used was Bamberg & Sternberg. I found that books treatment muddled and shortsighted. I mean, most of the linear algebra in the book deals only with 2 dimensional vector spaces. And the book was entirely useless in teaching differential forms...

So i went looking for a better book to learn diferential forms. i didn t like flanders, it was too brief. this is the book for me. Don t expect to find any linear algebra here, but you d better know lin. alg. before you open this book.

it is a challenging book, mathematically speaking, to study on your own (for a senior ugrad phys major, anyway), but it s treatment of forms and tensors is comprehensive, thorough, and detailed. and it shows you all the applications to relativity and electrodynamics, etc... it also builds up all the theory in with a background of differential geometry and topology, which are developed in the first chapter (but wasn t i glad to have already studied those topics beforehand!)

this book prepared me for my mathematical physics class, plus gave me months of other material to study. it is difficult, so i read and reread each chapter.

Bad book.
Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails.
Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al
2 stars for effort.

Good one, even if not the best, probably
This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.

There are better...
I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs. ... Read more

Book Description

This unique book explores the connections between the geometry of mappings and many important areas of modern mathematics such as Harmonic and non-linear Analysis, the theory of Partial Differential Equations, Conformal Geometry and Topology.Much of the book is new. It aims to provide students and researchers in many areas with a comprehensive and up to date account and an overview of the subject as a whole. ... Read more

Book Description

In the past few years elliptic curve cryptography has moved from a fringe activity to a major system in the commercial world. This timely work summarizes knowledge gathered at Hewlett-Packard over a number of years and explains the mathematics behind practical implementations of elliptic curve systems. Since the mathematics is advanced, a high barrier to entry exists for individuals and companies new to this technology. Hence, this book will be invaluable not only to mathematicians but also to engineers and computer scientists who want to actually implement such systems. ... Read more

Reviews (3)

too much math
This is a fairly complete treatment of elliptic curve cryptography. It suffers from a very uneven treatment. The chapters on implementation are well written and easy to read. The material on the logarithm problem, however, is much too advanced and will only be accessible to research mathematicians. A big omission in the book are protocols such as signatures and encryption.

Good compact book on elliptic curves in cryptography
This book gives a good summary of the current algorithms and methodologies employed in elliptic curve cryptography. The book is short (less than 200 pages), so most of the mathematical proofs of the main results are omitted. The authors instead concentrate on the mathematics needed to implement elliptic curve cryptography. The book is written for the reader with some experience in cryptography and one who has some background in the theory of elliptic curves. A reader coming to the field for the first time might find the reading difficult. The authors do give a brief summary in Chapter 1 on the idea of doing cryptography based on group theory. They then move on to discuss finite field arithmetic in Chapter 2. The reader is expected to know some of the basic notions of multiprecision arithmetic for integers. The authors choose to work with 2^16. Psuedocode is given for doing modular arithmetic with Montgomery arithmetic given special attention. The last section of the chapter gives a good summary of arithmetic in fields of characteristic 2. Chapter 3 discusses very compactly arithmetic in elliptic curves. This is where the reader should already have the background in the theory of elliptic curves, since the reading is very fast and formal. The authors do a good job of summarizing how modular polynomials come into play in elliptic curve cryptography and give some explicit examples of these polynomials. The most important chapter of the book is Chapter 4, where the authors give a discussion of how to implement elliptic curves efficiently in cryptosystems. This chapter is nicely written and pseudocode appears many times with lots of nice examples. This chapter serves as background for the next one on the discrete logarithm problem using elliptic curves over finite fields. The MOV attack, the anomalous attack, and the baby step/giant step methods are discussed very nicely. Random methods, such as the tame and wild kangaroo are discussed at the end of the chapter.

The next three chapters concentrate on how to actually generate elliptic curves for cryptosystems, with particular attention payed to the Schoof Algorithm. The chapter on Schoof's algorithm is more detailed than the rest of the chapters and this makes for better reading. The authors do discuss how to generate curves using complex multiplication although the discussion is somewhat hurried. The next chapter discusses how elliptic curves have been applied to other areas in cryptography, such as factoring, etc. A good discussion of the ECPP algorithm on proving primality ends the chapter. The authors end the chapter with a discussion of hyperelliptic cryptography. Anyone familiar with the theory of elliptic curves and how they are applied to cryptography will naturually ask if hyperelliptic curves have any advantages over the elliptic case. The authors never really address this explicity but do give examples on just what is involved in implementing hyperelliptic curves in cryptography. Overall a fine addition to the literature on elliptic curves in cryptography. One would hope that the authors would write a follow-up book on hyperelliptic curves and maybe on general algebraic curves and their possible use in this area.

Good book
I think this is one of the best introductions to elliptic curve cryptosystems. This book have all the last algorithms in the field. ... Read more

Book Description

No number has captured the attention and imaginations of people throughout the ages as much as the ratio of a circle's circumference to its diameter.With incisive historical insight and a refreshing sense of humor, David Blatner explores the many facets of pi and humankind's fascination with it-from the ancient Egyptians and Archimedes to Leonardo da Vinci and the modern-day Chudnovsky brothers, who have calculated pi to billions of digits with a homemade supercomputer. New in paperback. ... Read more

Reviews (28)

History of a Mystery
David Blatners book on the history of Pi is inspiring and concise. Going through the history, East and West, where the reader the will discover all the errors in formulating this most mysterious infinite number. Blatner also discusses how to remember the number, if need be, as well as some tidbits of information where pi has shown up, either in movies or books or wherever. The book essentially avoids mathematical equations, although they are displayed to those that understand them, the book is written so anyone can appreciate the the work involved in getting to pi. A fun read and reccomended as an introduction to finding the ratio of the circumference of a circle to its diameter of a circle.

Not great but also not bad...
"The Joy of Pi" is a short synopsis on the history of pi. Even if you are a math, engineering, life science, or economics major, there are bound to be details in this book that will enlighten you. I was somewhat disappointed in the lack of mathematical jargon in the book though. Also, the book is somewhat difficult to read due to the writing style of the author and the page setup of the book. Text is located in boxes or triangles or circles on nearly every page. It almost appears the author wanted to fill the space and decided to use an irregularly styled page format. Yet, if you are wanting something to read on your next flight or at the swimming pool, this could be a book for you. If the subject matter interests you enough, you might find that there are no references to pursue this information further. This is a drawback to the book. Equations containing pi are included in the book but it is up to you to seek out the proof for these equations.

A little too easy
Though the subject is very attractive, but i think it's too easy for a normal-educated person who loves maths or natural science. If you want tp buy a book of this kind, "the history of pi" will do.

Mostly a Good Review of Pi and its history
Unique and eye catching format. The author, however, does seem to dismiss the fact that tht Bible discovered Pi and did so quite accurately before most else. He seems to do so for no good reason. He neglects that the Bible commonly rounds numbers (this isn't a blueprint!). And the Hebrew mechanics itself do reveal a percise value for pi (3.14150943...) This is only a difference of 0.0000832 with actual pi, making the Bible's description of pi the most accurate in antiquity! The author embarasses himself and reveals is lack of research when he calls this "dubious."

3.141592.........
I enjoyed this book immensely. It is neither too long to be ponderous for its subject - a non-technical review of the fascination of the number pi - nor too brief to be trivial. Its graphic nature and content went a long way to maintaining my interest and stimulating my own ideas. Maybe I do have a way to square the circle!

Just kidding!! ... Read more

Book Description

This book begins with a new approach to the geometry of pseudo-Finsler manifolds. It also discusses the geometry of pseudo-Finsler manifolds and presents a comparison between the induced and the intrinsic Finsler connections. The Cartan, Berwald, and Rund connections are all investigated. Included also is the study of totally geodesic and other special submanifolds such as curves, surfaces, and hypersurfaces.Audience: The book will be of interest to researchers working on pseudo-Finsler geometry in general, and on pseudo-Finsler submanifolds in particular. ... Read more

Book Description

This updated edition will continue to meet the needs for an authoritative comprehensive analysis of the rapidly growing field of basic hypergeometric series, or q-series.It includes deductive proofs, exercises, and useful appendices. Three new chapters have been added to this edition covering q-series in two and more variables; linear- and bilinear-generating functions for basic orthogonal polynomials; and summation and transformation formulas for elliptic hypergeometric series. In addition, the text and bibliography have been expanded to reflect recent developments. First Edition Hb (1990): 0-521-35049-2 ... Read more

Book Description

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. "Rational Points on Elliptic Curves" streses this interplay as it develops the basic theory, thereby providing an opportunity for advance undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum.This accessibility, the informal writing style, and a wealth of exercises make "Rational Points on Elliptic Curves" an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. ... Read more

Reviews (2)

Full of useful information and a great guide to intuition
The authors do a fantastic job of introducing elliptic curves for individuals and students interested in this area. Because of the importance of elliptic curves to cryptography, in integrable models in statistical mechanics, in superstring theory in physics, in mirror symmetry in algebraic geometry, in mechanics in the solution of the spinning top, and even in financial engineering, this book will be useful in building intuition about these interesting objects. Be careful in reading this book though...the theory of elliptic curves is beautiful and addicting, and you will want no doubt to read more about them after finishing it. There are two other books by Silverman that will alleviate the monkey on your back for more knowledge about elliptic curves. Happy reading......

Excellent Introduction to Elliptic Curves
This wonderful book is an excellent introduction to elliptic curves over the rational numbers. It is self-contained and easily accessible, but still takes the reader quite far, thus giving an undergraduate reader some exciting glimpes of deeper mathematics. This book is ideally suited as a text book for an undergraduate course (I have myself enjoyed it as a course), but is written in a lively style that also makes it fun to read on one's own. It covers such topics as the Nagell-Lutz Theorem, Mordell's Theorem over rational numbers, elliptic curves over finite fields and reduction modulo p, Thue's Theorem and diophantine approximation, and even an introduction to complex multiplication. An appendix provides the reader with a basic background on projective geometry. This book is a must for any student wanting to see beyond the ordinary coursework, and at the same time provides a natural stepping stone to a more advanced treatment of the subject, such as "The Arithmetic of Elliptic Curves", also by Silverman, which has become pretty much the standard text on the subject. ... Read more

Book Description

Many secondary mathematics teachers and teacher educators feel ill-equipped to teach a course in geometry. Most feel fortunate to stay one chapter ahead of the very students they are teaching! Most college-level geometry texts don't address their specific needs. Advanced Euclidian Geometry was written to fill this void in many ways: • Provides both in-service and pre-service teachers with a solid review of the essentials of the high school geometry course - AND MORE! • Offers further elaboration of those concepts (advanced Euclidian geometry, the 19th and 20th century expansions of Euclid's work) to give teachers more confidence in guiding student explorations and answering questions. • Contains hundreds of illustrations created in The Geometer's Sketchpad(r) Dynamic Geometry(tm) software. • Is packaged with a dual (Windows(r)/Macintosh(r)) platform CD-ROM containing over 100 interactive sketches using The Geometer's Sketchpad (assumes that users has access to the Sketchpad program) ... Read more

Reviews (1)

A must-have book for every geometry teacher
As a high-school geometry teacher, I have often wished for a book like this. Sadly, the typical one-year geometry course comes to an end just as students are within reach of some truly beautiful and intriguing theorems. Dr. Posamentier's book begins where most high-school geometry textbooks end, and presents many wonderful results that lie just beyond their purview: the nine-
point circle; the golden rectangle; the theorems of Ceva, Menelaus, Ptolemy, Pascal, Desargues, and Brianchon; excircles and incircles; cyclic quadrilaterals; and much more. This book provides a rich geometric feast.

There are several books that cover much of the same ground that Dr. Posamentier surveys: GEOMETRY REVISITED, by Coxeter and Greitzer, and EPISODES IN NINETEENTH AND TWENTIETH CENTURY EUCLIDEAN GEOMETRY, by Honsberger, are two of the best. What makes Dr. Posamentier's book stand out is its usefulness as a textbook. Theorems are fully proved, and arranged in a logically coherent sequence. (The book is Euclidean in format as well as in subject matter.)

The book is thoughtfully designed. The pages are large, the type is easy to read, the diagrams are clear, and the book lies flat when opened.

EVERY HIGH-SCHOOL GEOMETRY TEACHER SHOULD HAVE THIS BOOK. It's a rich source of supplementary material for regular sections, and an ideal textbook for the second semester of an honors-level class, or for a student who wants to pursue the study of geometry on an independent-study basis.

I know that I'll be turning to this book again and again next year, and for as long as I teach geometry. In fact, I plan to buy another copy so that I'll have one at home and one in my classroom.

Bravo, Dr. Posamentier, and thank you! ... Read more