Book Description

Following on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals.

Much of the material presented in this book has come to the fore in recent years. This includes methods for studying dimensions and other parameters of fractal sets and measures, as well as more sophisticated techniques such as thermodynamic formalism and tangent measures. In addition to general theory, many examples and applications are described, in areas such as differential equations and harmonic analysis.

This book is mathematically precise, but aims to give an intuitive feel for the subject, with underlying concepts described in a clear and accessible manner. The reader is assumed to be familiar with material from Fractal Geometry, but the main ideas and notation are reviewed in the first two chapters. Each chapter ends with brief notes on the development and current state of the subject. Exercises are included to reinforce the concepts.

The author's clear style and up-to-date coverage of the subject make this book essential reading for all those who with to develop their understanding of fractal geometry. ... Read more

Reviews (2)

Dimension of fractal objects
A suitable book to remove any doubt about calculation of dimension of fractal objects. I enjoyed the chapter about ergodic theorem.

Maybe this book is advanced version of "Fractal Geometry"
I had read "Fractal Geometry" in last year. Then I purchase this book. It seems advanced version of "Fractal Geometry". In this book, some applications of fractal for science and engineering. For example, thermodynamic formalism, ergodic theorem, multifractal analysis, differential equations, and so on. I think that this book will become good textbook for scientist and engineer who apply fractal geometry for their field. ... Read more

Book Description

This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research.The first part, which grew out of Tate's Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the rationals.The next two chapters recast the arguments used in the proof of the Mordell theorem into the context of Galois cohomology and descent theory.This is followed by three chapters on the analytic theory of elliptic curves, including such topics as elliptic functions, theta functions, and modular functions.Next, the theory of endomorphisms and elliptic curves over infinite and local fields are discussed.The book then continues by providing a survey of results in the arithmetic theory, especially those related to the conjecture of the Birch and Swinnerton-Dyer.

This new edition contains three new chapters which explore recent directions and extensions of the theory of elliptic curves and the addition of two new appendices.The first appendix, written by Stefan Theisan, examines the role of Calabi-Yau manifolds in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory.

Dale Husemöller is a member of the faculty at the Max Planck Institute of Mathematics in Bonn. ... Read more

Reviews (1)

An excellent update to the first edition.
Anyone who has studied elliptic curves appreciates their beauty and the richness of the mathematics that arises from such a study. This book, first published in 1987, has three additional chapters that reflect some major applications of elliptic curves since then. Indeed, the resolution of Fermat's Last Theorem due to Andrew Wiles and the use of a generalization of elliptic curves, called Calabi-Yau manifolds, in string theory have all taken place since the time of publication. The review here will be confined to these chapters.

Chapter 18 is a brief summary of the modular elliptic curves conjecture and Fermat's Last Theorem from mostly an historical perspective. The author reviews the material from prior chapters that relate to the modular curve conjecture. The Tate module of an elliptic curve plays a central role, with its structure as an l-adic Galois module allowing the author to formulate an alternative version of the modular curve conjecture. The author shows that the modular elliptic curve conjecture is equivalent to the assertion that every l-adic representation arising from a Tate module of an elliptic curve over the rational numbers Q comes from a modular form of weight 2, which is a Hecke eigenfunction.

It is fascinating that the connection between elliptic curves and Fermat's Last Theorem was only pointed out as late as 1986 by the mathematician Gerhard Frey. The relation of the 'Frey curve', as it is now called, to Fermat's Last Theorem is discussed by the author, and he shows how it is reduced to the modular elliptic curve conjecture for semistable curves.

In chapter 19, the author introduces the reader to Calabi-Yau varieties, which are higher dimensional analogs of elliptic curves, and which have become very important in high-energy physics. The reader will have to have some background in the theory of complex manifolds to appreciate this chapter, but the author does a quick survey of the relevant topics. Of particular importance in this discussion are the Kahler manifolds, which can be thought of as complex manifolds with a metric that is an analog of the Euclidean metric in the real case, i.e. the metric is Hermitian and is closed.

After a further review of characteristic classes the author gives several equivalent definitions of Calabi-Yau manifolds, and several examples in (complex) dimension one, two, and three. He also gives examples of Calabi-Yau manifolds that arise from projective and weighted projective spaces, and their generalizations, the toric varieties. A brief remark is made concerning the existence of 'mirror' Calabi-Yau manifolds, these latter objects currently the subject of intense research. Just as in the case of real manifolds, it is of interest to find invariants for Calabi-Yau manifolds that will assist in their classification. The author does this for the case of surfaces that are Calabi-Yau, and this naturally leads to the analog of the Euler characteristic in the guise of the famous Riemann-Roch theorem. The Riemann-Roch theorem though is not proven, but the author does show explicitly how to obtain the formula for the genus for the structure sheaf on the scheme defined by the ideal sheaf. A brief introduction to K3 surfaces is given. These surfaces are very important in physical applications and in four-dimensional topology.

Finally, in the last chapter of the book, the author studies families of elliptic curves. This is done in the context of the theory of schemes, and the author makes some connections with physics. The author gives a very brief review of scheme theory, starting with the notion of a 'local ringed space', which is a topological space with a sheaf of rings defined on it such that the stalks are local rings for every point in the space. Local ringed spaces include smooth and complex analytic manifolds as special cases, and codify both the algebraic and analytic properties of the objects studied. An affine scheme is then defined as a locally ringed space isomorphic to the spectrum of a ring. A scheme is a locally ringed space locally isomorphic at each point to an affine scheme. The isomorphism classes of elliptic curves have the structure of a scheme.

Elliptic fibrations of surfaces over curves are studied in terms of their effective divisors, which are analogs of the canonical divisors used in the Enriques classification of surfaces. The Euler characteristic is then computed in terms of the effective divisor. The author then shows that a K3 surface with a Picard number at least 5 has an elliptic fibration. This is generalized to the case of Calabi-Yau varieties using the concept of a 'numerically effective' divisor. Some explicit examples of Calabi-Yau hypersurfaces in four-dimensional weighted projective are then given. These examples were found by string theorists, and the author therefore devotes an appendix that describes how Calabi-Yau manifolds are used in high energy physics. The appendix is very short, and a perusal of the literature of string theory will reveal the overwhelming importance of Calabi-Yau manifolds. String theory has evolved into M-theories and membrane theories, but both of these involve heavy use of algebraic geometry, and many of the constructions are generalizations of what is known for the case of elliptic curves. ... Read more

Book Description

Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections. ... Read more

Book Description

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students.Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired.By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory. ... Read more

Reviews (1)

Excellent
Anyone who writes a book on elliptic curves will never do a bad job, for these objects are so beautiful that it would be a sacrilege to do otherwise. Those who study elliptic curves fall under their spell, not only because of their beauty, but also because of their many applications: the spinning top in mechanics, cryptography, exactly solved models in statistical mechanics, precession of the Mercury perihelion in general relativity, the proof of Fermat's Last (Wiles) Theorem, control theory, and string theory, to name a few. This book is an excellent treatment of ECs and would be good for a graduate student starting out in the field. The author gives many concrete examples of the main theorems, and helpful exercises are found at the end of each chapter.

The author begins the book with two neat problems that motivate well the subject of elliptic curves: the pyramid of cannonballs and the right triangle problem, i.e. which integers can occur as areas of right triangles with integer sides? He then immediately begins the elementary theory of ECs in chapter 2. The treatment is pretty standard, although he proves Pascal's and Pappus's theorems using the associativity of the group operation on ECs, which is not usually done in books on ECs. Also somewhat non-standard this early in the game is the discussion of reduction of ECs modulo various primes, and the subsequent definitions of additive, split multiplicative, and non-split multiplicative reduction.

The study of torsion points is done in chapter 3 with the Weil pairing on the n-torsion of an EC taking center stage. A fairly short chapter, the author delays the proof of the properties of the Weil pairing until chapter 11, where it is done with divisors.

Chapter 4 deals with elliptic curves over finite fields, and is one of the most important in the book from the standpoint of cryptographic applications of ECs. Hasse's theorem, giving the bounds for the group of points on an EC over a finite field, is proven in detail. The Frobenius endomorphism is introduced, and a proof of Schoof's algorithm for computing the number of points on ECs over a finite field is given a detailed treatment. There are many symbolic computational software packages in both the open and commerical realm which will do the counting straightforwardly, and anyone interested in cryptography will need to be familiar with some of these. Supersingular curves in characteristic p are introduced, and the author gives a good discussion of the reason why they are named as such.

The discrete logarithm problem, a topic also very important for cryptographic applications, is discussed in chapter 5. The chapter beings with the index calculus, and, recognizing that it does not apply to general groups, the Pohlig-Hellman, baby step-giant step method, and Pollards rho and lambda methods are discussed in details. The author then shows that for supersingular and "anomalous" curves, that the discrete logarithm problem can be reduced to an easier discrete logarithm problem. Along the way, two important concepts are introduced: the p-adic valuation, and the Tate-Lichtenbaum pairing, the latter of which is related to the Weil pairing, but applies to situations where the Weil pairing does not.

Elliptic curve cryptography is then discussed in chapter 6, and the treatment is fairly thorough. The author shows to what extent the Decision Diffie-Hellman problem can be solved using the Weil pairing. He also shows how to represent a message on an elliptic curve, satisfying early on any reader's curiosity on just how this is done. The El Gamal and ECDSA are compared in terms of their computational efficiency. An EC generalization of RSA is also discussed in some detail, along with a cryptosystem based on the Weil pairing. Chapter 7 then gives other applications of ECs, such as factoring and primality testing.

Chapter 8 marks the beginning of the "heavy artillery" in the theory of ECs, for here the author begins the discussion of elliptic curves over the rational numbers, which can be viewed as an example of Diophantine geometry. The famous Mordell-Weil theorem is proved, and as a sign that one is definitely in the arena of modern mathematics, the proof is given in terms of Galois cohomology, which is an abstraction of the Fermat method of descent. The reader gets a taste of height functions, and via some good examples, gets insight into why the rank of the EC is so difficult to compute. A neat example is given of a nontrivial Shafarevich-Tate group.

I did not read the chapters 9, 10, or 11 on ECs over the complex numbers, complex multiplication, and divisors, so I will omit their review. Chapter 12 introduces the famous zeta functions, and their use in obtaining arithmetic information about an EC. Zeta functions motivate the definition of an L-function of an EC, these being tremendously important in modern developments in the theory of ECs, such as the Swinnerton-Dyer and Birch conjecture, the latter of which is motivated rather nicely in this chapter.

The last chapter of the book is an excellent introduction to the proof of Fermat's Last Theorem. Considering the level of the book, the author captures very well the essential ideas. Readers will be well prepared, after studying more algebraic number theory and the theory of Galois representations (which the author only skims in the book), to tackle the full proof if so desired. ... Read more

Reviews (1)

Excellent Electrical Engineering Reference Book
I recently tutored a co-worker for the electrical engineering Professional Engineering license exam and this book greatly helped out in explaining Laplace transforms. The book covers transform circuit analysis by starting out with the basics and then building up to Laplace transforms. After reading this book and working through the examples, I feel much stronger in this area of electrical engineering. I have a B.S. in electrical engineering and a P.E. license since 1995. I have read several other books on this subject, including differential equations text books and I think this book presents the clearest explanations and examples applied directly to electrical engineering. I would recommend this book to anyone with an interest in electrical engineering. ... Read more

Amazon.com

Bernhard Riemann was an underdog of sorts, a malnourished son of aparson who grew up to be the author of one of mathematics' greatestproblems. In Prime Obsession, John Derbyshire deals brilliantlywith both Riemann's life and that problem:proof of the conjecture,"All non-trivial zeros of the zeta function have real part one-half."Though the statement itself passes as nonsense to anyone but amathematician, Derbyshire walks readers through the decades of reasoningthat led to the Riemann Hypothesis in such a way as to clear it upperfectly. Riemann himself never proved the statement, and it remainsunsolved to this day. Prime Obsession offers alternating chaptersof step-by-step math and a history of 19th-century European intellectuallife, letting readers take a breather between chunks of well-writteninformation. Derbyshire's style is accessible but not dumbed-down,thorough but not heavy-handed. This is among the best popular treatmentsof an obscure mathematical idea, inviting readers to explore the theorywithout insisting on page after page of formulae.

Reviews (38)

An Excellent Read, Highly Recommended
Prime Obsession is an excellent popularization of the Riemann Hypothesis. I found John Derbyshire's presentation of the math to be very approachable by non-mathematicians like myself. It's taken slow, one basic step at a time, and spread across a well written and fascinating history of Bernhard Riemann and other key players. Simply put, you do not need an advance degree in mathematics to enjoy this book.

My math bakground is limited to 2 semesters of calculus 20 years ago and I haven't used it since. For me, John Derbyshire's approach was both refreshing and entertaining. If you've got even the faintest interest in math, you will find this book rewarding.

Read this one for the pure entertainment value of it all.
I found this to be a rather delightful book with its arrangement of chapters alternating between historical point of view back to mathematical progress and then back to historical.

I found it very entertaining to read about the lives of the great mathematicians involved in developing the prime number theory and furthering the study of the Riemann Hypothesis. Mathematics is littered with such interesting characters that even a liberal arts major can enjoy these expository stories of their lives.

The only downside to this whole book is that he takes too much time for the non-math inclined readers to get 'caught up' with their basic skills before he jumps to anything interesting. If you have a background that is strong through calculus, then you could probably avoid reading all the math-based chapters up through the end of the prime number theory section of the book, and you most likely woud not have missed a thing.

Complex Math Made Very Understandable and Interesting
Although this book deals with a subject that no-one would sensibly place in a category below "Very Advanced," John Derbyshire treats his subject as well as any math author I've ever read, and I've read a lot of math books over the past 40-some years.

My formal math education ended after a standard introductory calculus course as an undergrad. However, I have always been, and remain, extremely interested in math -- a math aficianado if you will. As such, I've self-taught myself a lot of math -- including a lot of very advanced math -- over the past 40 years; ergo, my reading of a great many math books. And without doubt, Derbyshire's book is the finest math book I've yet to read.

I suspect Derbyshire started with the hypothesis that his readers are not familiar (or only familiar in a passing sense) with high-level, advanced math, and perhaps might even suffer from math anxiety. Any such readers, however, should have absolutely no fears. Derbyshire's exposition is superb. He clearly defines everything the reader needs to know to grasp AND understand fully the more advanced parts of the book. The book is clearly well designed to convey the information he wants or needs of convey and masterfully explains what would otherwise be quite difficult to understand.

Without any doubt this is by far the best book on any advanced and complicated subject -- the best book on ANY math subject (including a book on something as simple as how to add one and one) -- I have ever read.

Without sacrificing the complexity of the subject, Derbyshire has written his book in a very readable and interesting manner. And he does all this while making the subject so interesting you can hardly wait for someone to finally prove Riemann's Hypothesis and Riemann's zeta function so we can read Derbyshire's account of that landmark event in the history of mathematics.

splendid (though heavy math)
This book should be the first one to appear in Amazon's listings for the Riemann Hypothesis, yet doesn't even appear in the top ten. It gives fascinating historical background to a very real Riemann and his friends, traces developments to the present day in a conversational tone, and somehow manages to take the reader through the details of what the RH says so that you actually understand it. Recommended with one reservation; to understand the chapters (every other one) which bring one to understand the RH, you will need to make a considerable investment in reading and rereading to make it. That is not for the faint of heart. However, the other half of the book can be enjoyed by anyone who likes general science history books.

What a piece of work a man is!
"Prime Obsession" is a fascinating book for several reasons: the author explains a difficult topic with such clarity that it's simply amazing. For those who are more skilled in math, this book would also be very enjoyable to read, except that they might find some of his explanations redundant because he really assumes that the we don't know anything (and I mean anything!).
Mr. Derbyshire obviously understands the topic quite well himself. He has written an amazing book for everyone to enjoy.
200 years since Riemann first presented the problem, we are still desperately trying to solve it, and one day, you never know... what a piece of work a man is! ... Read more

Book Description

Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris.After receiving his Ph.D. from Princeton in 1963, Hartshorne became a Junior Fellow at Harvard, then taught there for several years.In 1972 he moved to California where he is now Professor at the University of California at Berkeley.He is the author of "Residues and Duality" (1966), "Foundations of Projective Geometry (1968), "Ample Subvarieties of Algebraic Varieties" (1970), and numerous research titles.His current research interest is the geometry of projective varieties and vector bundles. He has been a visiting professor at the College de France and at Kyoto University, where he gave lectures in French and in Japanese, respectively.Professor Hartshorne is married to Edie Churchill, educator and psychotherapist, and has two sons.He has travelled widely, speaks several foreign languages, and is an experienced mountain climber. He is also an accomplished amateur musician: he has played the flute for many years, and during his last visit to Kyoto he began studying the shakuhachi. ... Read more

Reviews (5)

THE book for the Grothendieck approach
This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.

Some helpful suggestions from my experience with this book:
1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes;
2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.

Experiences of a rather below-average graduate student
(that's me.)

I agree with the other reviewers' comments concerning the phenomenal depth and breadth of the topics covered in this book. Hartshorne builds the soaring edifice of modern algebraic geometry from the ground up. All the way through, the exposition is concise and absolutely clear. The proofs strike an excellent balance between meticulousness and readability.

The approach he takes seems to be to try to acquaint the reader with as much formalism as possible as quickly as possible, and he seems reluctant to offer any sneak previews of vital concepts such as divisors, differentials, and flatness until the reader's brain is "ripe". As a result, Hartshorne is able to state and prove results under extremely general hypotheses. This approach also benefits the kind of reader who wishes to use this as a reference book.

It's important also to note the disadvantages of Hartshorne's approach: Time and again, I found myself utterly baffled by the definitions, because the motivations for them are lacking.

To give a minor example, take the definition (in chapter 1, part 3) of a morphism between two varietes. First, regular functions from a variety over k to k are defined as those that are locally representable as quotients of polynomials (without bothering to give an example of a case of a regular function for which more than one such representation is needed). Then a morphism f: X -> Y is defined as a Zariski-continuous function with the property that whenever you have an open subset V of Y, and a regular function V -> k, then f^-1(V) -> V -> k is regular. There's nothing wrong with this definition, of course, but I found it very difficult to make sense of, initially. A morphism, after all, is supposed to be something that preserves structure, but it's not immediately obvious what "structure" is being preserved in this case (and the full details of this aren't spelt out until much later, after sheaves have been defined). A better didactic approach, I think, would be either (1) to define morphisms of affine varieties simply as functions given by polynomials, and then show that the above definition is the only natural way of generalising this, or (2) to briefly introduce sheaves at the outset, making it clear that the "structure" we wish to define on a variety consists precisely of the sheaf of regular functions.

Another negative effect of Hartshorne's approach is that, if you have to traverse a mire of formalism before meeting an idea, it makes the idea seem more complicated than it actually is.

Certainly there's nothing to stop a dedicated reader just ignoring any temporary befuddlements, secure in the knowledge that eventually everything will make sense, but not all of us have the patience. This book contains an almost ridiculous number of exercises - most of which are supposed to be "formalities", there to flesh out the definitions, but many contain absolutely crucial definitions and lemmas. Attempting to do all the exercises as you go along is very taxing work indeed, and becomes demoralising whenever you get stuck. Perhaps the best strategy is to do only those exercises that are interesting or important for later work.
Also, as others have noted, this book is very tough going on those who don't already have some familiarity with commutative algebra and (later on) homological algebra.

All in all, I think this book will be most useful for people who already know quite a lot of algebraic geometry, commutative/homological algebra etc., and are wishing to consolidate and "modernise" their understanding. For beginners, it's a struggle, but not an unproductive one, especially if assisted by other, less demanding books.

Be prepared...
This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century.

Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.

The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.

The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.

Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.

This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.

Terrific, if you want it.
This book hardly needs a review on Amazon, because if you have as much math background as it needs, then you must already know it is indispensible for learning about schemes in algebraic geometry. The book is clear, concise, very well organized, and very long. If you do not already know the Noether normalization theorem, and the Hilbert Nullstellensatz, then you do not want this book yet--you want an introduction to commutative algebra.

Indispensable!
Excelent and useful text, indispensable for graduate students and research ,athematicians working on algebraic geometry. Hartshorne walks the fine line between commutative algebra and their geometrical counterparts with elegance. The book is also rich in references, providing many directions for further study. ... Read more

Book Description

This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics.They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis. ... Read more

Reviews (2)

A very nice little book
This is a very nice, short introduction to the subject -- This series of blue paperbacks by CUP is excellent. Typically, all books in the series are readable introductions. Somewhat higher level than the corresponding series from Springer (the one where all exercises have full solutions).

Incidentally, the author is a very attractive woman.

Well suited as an introduction to algebraic curves
The book gives a good general overview of algebraic curves using only elementary algebra, topology, and complex analysis. There are lots of diagrams of elliptic curves in the historical introduction in the first chapter and the subject is well motivated. Hilbert's Nullstellensatz is introduced in the context of real algebraic curves as an answer to the question of when the polynomials definte the same curve. The visualization approach taken by the author in the first chapter has taken on dramatic proportions do to the computer graphics packages currently available. The author introduces complex algebraic curves in complex 2-dimensional space in the next chapter. Recognizing that such curves are not compact, he compactifies them by adding suitable points at infinity, giving complex projective curves. The algebraic properties of these curves are studied in the next chapter. He does a good job of motivating the group law on elliptic curves on the last theorem of the chapter, leaving the proof of associativity to the reader in the exercises. The topology of complex projective curves is taken up in Chapter 4. The author gives two proofs of the degree-genus formula, one geometric and the other from a holomorphic point of view. This leads to a consideration of branch points and ramified covers. The author's outline of the proofs is very detailed and therefore very helpful to one encountering the proof for the first time. The statement of the formula via the Riemann-Roch theorem in more formal treatments (and later in the book) can then be appreciated more. The subject of non-singular complex projective curves, namely Riemann surfaces, is effectively discussed in Chapter 5, with holomorphic differentials outlined in Chapter 6. The Riemann-Roch theorem makes its appearance here, and the author is careful to point out its use as an alternative characterization of the genus given earlier by topological arguments. Divisors are introduced as formal sums, but their understanding is straightforward here because the author has motivated them with a discussion of the properties of holomorphic and meromorphic functions earlier in the chapter. The proof of the Riemann-Roch theorem is very detailed and understandable. The book ends with a discussion of singular curves via resolution of singularities. Newton polygons and Puiseux expansions are used to investigate the behavior of degree d projective curves near a singular point. The geometrical constructions used here by the author are of great help in understanding the behavior of these curves. A very well-written book for students and new-comers to the area of algebraic curves. It will pave the way for more advanced reading on the subject. ... Read more

Book Description

This volume is the second edition of the highly successful Fractals Everywhere.The Focus of this text is how fractal geometry can be used to model real objects in the physical world.

This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.

Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software.The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry

* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.
* Problems and tools emphasizing fractal applciations.
* An all-new answer key to problems in the text, with solutions and hints. ... Read more

Reviews (9)

Sometimes annoying but instructive
Although instructive, this book is sometimes annoying to read. The author seems to be playing his cards very close to the vest and not telling us everything.

For instance, there is little or no instruction on how to implement the IFS attractors presented as a panacea for data compression. This seems to be proprietary to his company. It also seems that hands-on manipulation is crucial to the images' production, contrary to the author's claims.

If you can understand the mathematics you may find the book useful, as I did when writing my book Fractals in MUsic.

A bad book for 7 th graders like me
this is a bad and very confusing book for a young student in, say... 7th grade, like me. The language is incomprehensible and there are no visual aids.

Opinión general
HUmmm!! parece interesante este librito. Pero la verdad busco uno donde encuentre aplicaciones a la ingeniería.
Estos libros de teoría suelen ponerse aburridos al no tener sufuciente información sobre aplicaciones.
De todos modos apenas lo tenga en las manos y lo mire doy una opinión más seria de este.

Good book for applications of fractal geometry, but....
This is a good book on applications of fractals; the chapters on modeling natural objects with iterated function systems (IFS) and fractal interpolation are especially useful. Many standard topics are included, for example, fractal dimension, Julia and Mandelbrot sets, chaos and the shift dynamical system. Some of the illustrations are captivating.

However, the book is not well organized, and the writing is extremely wordy to the point of being irritating. Some paragraphs read as if they belonged to a "Dummies" handbook. Also, I have to agree with one reviewer that the treatment of fractal dimension is poor. For one thing, it does not fully develop the intuition behind the concept-- much less the math. This same remark holds for the chapter on chaotic dynamics.

In summary, the book is fine for applications, but supplement your reading with a more substantial text.

Good computer graphics book. Bad mathematics book.
This book was written by a regarded expert in the fields of digital image processing and data compression, and illustrates well how some "abstract" mathematical concepts can be applied successfully to such purposes.

However, it is evident that it was written in a rush, and the results can be seen. I have found a lot of typographic mistakes, errors in the exercises, and even errors in some of its mathematical proofs. Also, the author pays almost no attention to the fundamental concept of fractal geometry: the fractal dimension.

I read this book because I needed a strong background in fractal geometry to write my Bachelor's thesis, but got dissapointed because of its mathematical defficiencies, and eventually decided to move to better sources on the subject.

Please check my other reviews in my member page (just click on my name above). ... Read more

Book Description

The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study. ... Read more

Reviews (8)

Objects and maps are everywhere
Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.

Very uneven, but still useful
As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts.

Heavy Hitter Strikes Out
I sure hope Schanuel wrote this book and the publisher simply tacked on
Lawvere's name for marketing purposes. This text is a fantastic
example of why research mathematicians should not write for John Q.
Public. The random, pointless examples scattered throughout the book
remind me of the "word problems" that were so popular in high school
algebra texts written after the Chicago School hijacked the educational
textbook market.

After teasing the reader with examples of real mathematics, e.g.
Pick's Formula, the authors stop short of actually proving a theorem
and scurry back to their shelter of objects and arrows where they can
safely field trivial questions by ersatz students with politically
correct names.

Perhaps Category Theory is just not something that is accessible to the
general public? High school math teachers (I assume one intended
audience for the text) that can achieve even the slightest appreciation
of why Eilenberg and Mac Lane invented Category Theory are surely as
rare as rocking-horse poop.

What I would really like to see from someone as eminent as Lawvere write a
first year graduate level book that covers elementary set theory and/or
logic using Category Theory. Translating Model Theory and Topoi(1.) to
this level would be a good start. College math professors are really
the only people in a position to understand and transmit this beautiful
theory to aspiring mathematicians.

1. Model Theory and Topoi, Lecture Notes in Mathematics 445,
Springer-Verlag 1975

Keith A. Lewis ...

A retract in search of a section
There is a wonderful course in category theory for high school students, just begging to be excavated from this multi-layered book.
Please don't be put off by the disjointed and uneasy combination of materials that cluster around certain themes. You know you will have a lot of work to do when the same definition (of monomorphism) is presented both on page 52 and also on page 336.
With all the elementary themes covered in many varying ways, it would be best to consider this book as having been structured as a retract for which your job will be to construct the appropriate section.

A Good Introduction
As a first introduction to Categories, this book is well written, clever, simple and very clear. However, I was disappointed with it. From the notoriety of the authors and the, yes, cool illustrations I assumed it would be a gem. However, it fell short. I've been toying with Category Theory for a few years, and every time I try to get into a book on Categories I get stumped at the notions of Functors and Natural Transformations. This book, however, dealt with neither at length, despite the fact that Category Theory originated around the notion of Natural Transformations in the first place. (As I understand it at least.) That said, there are many very cool passages in the book, including a functional analysis of a Chinese restaurant and an elegent exposition of Brouwer's Fixed Point Theorem.

Still, for my purposes, I prefer Robert Goldblatt's "Topoi: The Categorical Analysis of Logig" and Michael Barr's "Category Theory for Computing Science". As both are intended for non Category Theorists, both build their presentations of Category Theory from sratch. Sadly, I think both are out of print. Not for the faint of heart, I'm told Saunders Mac Lane's "Categories for the Working Mathematician" is the classic. (It's on my wish list.) ... Read more

Book Description

This book is meant to be an introductory text, albeit at an upper graduate level. Themain prerequisite for reading this book is some familiarity with the basic theory of ellipticcurves as described, for example, in the first volume. Numerous exercises have beenincluded at the end of each chapter. A list of comments and citations for the exerciseswill be found at the end of the book. ... Read more

Reviews (1)

The most fascinating objects in all of mathematics
This book is a continuation of the authors earlier book on elliptic curves, which was also an excellent book, and treats the more specialized topics in elliptic curves. I cannot think of a branch of physics or engineering that has not made use of some facet of the theory of elliptic curves, and they have myriads of applications in other fields also, such as cryptography and financial engineering. The book is very organized, straightforward to read, the author summarizes well his intentions at the beginning of each chapter, and recommends several references for topics left out of the main discussion. Space does not allow a detailed chapter by chapter review, so I will confine my review to the first two chapters, which were of main interest to me. In summary, Chapter 1 discusses how to study elliptic curves by taking a collection of them, each member being isomorphic, and studying the properties of modular functions and differential forms on this collection, now thought of as an algebraic curve, called the moduli space. The famous linear operators, called the Hecke operators, act on the the space of modular forms, and they and their eigenfunctions satisfy the same set of relations. One then attaches the well-known L-series to the modular forms that has very interesting algebraic and analytic properties. In more detail, the author does the following in the chapter. The set of lattices in the complex plane modulo non-zero multiplication L/C* is considered, along with the set of elliptic curves over the complex plane modulo complex isomorphism. These collections are proven to be bijective by showing that L/C* is isomorphic to C by first putting a complex structure on it. This leads to a surjective map from the upper-half plane H to L/C*. Proving this to be injective leads to a bijection from SL2(Z)\H to L/C*. Since the matrix -1 acts trivially on H, one can quotient out +1 and -1 and obtain the modular group. The quotient space modular group\H is a 2-sphere minus a point, but can be made into a Riemann surface by extending the upper half-plane (called H*). The modular curve X(1) = modular group\H* results and is compact and Hausdorff. A complex structure is put on it, making it into a a Riemann surface of genus 0. Meromorphic functions on X(1) are rational functions of the j function, but more interesting functions are defined on X(1), namely the modular functions, such as the Eisenstein series. These considerations lead to a proof of the uniformization theorem for elliptic curves over C. For a given elliptic curve E, a study of the set of all isogenies to E of degree n is the same as that of studying degree n maps from E to other elliptic curves, which is called the dual isogeny, and leads to the Hecke operator. The Hecke operator and the homothety operator both map the divisor group of the lattice to itself, and generate a commutative algebra, called the Hecke algebra. Hecke operators can act on modular forms of weight 2k, and modular forms exist which are simultaneous eigenfunctions for the Hecke operator of weight 2k. It can be proven, but the author does not do so, that the normalized eigenfunctions form a basis for the space of cusp forms of weight 2k. The Fourier coefficients of the eigenfunction have an Euler product decomposition of a Dirichlet series attached to f, called the L-series. In the next chapter, the author considers elliptic curves that have extra endomorphisms, called complex multiplication. The collection of endomorphisms is usually taken to be the real numbers R, or R(K), which is the ring of integers of R tensored with the rational numbers. And, just as in chapter 1, he studies collections of elliptic curves, but here ones with the same endomorphism ring., called ELL(R) in the book. Asking the question of how to construct an elliptic curve with complex multiplication by a particular R(K) leads him to studying the ideal class group of R(K), and this group is shown to act transitively on ELL(R(K)). The author also shows that every elliptic curve with complex multiplication is defined over an algebraic extension of Q. Several interesting examples of ellipti curves with complex multiplication are given. After a brief review of class field theory, the author proves that K(j(E)) is the Hilbert class field H and shows how the Galois group of H/K acts on j(E). The torsion points of E are then used to generate abelian extensions of K, using the Weber function for E/H, thus generalizing the usual cyclotomic extensions of number theory. Very interesting examples are given of these constructions and it is also shown that j(E) is an algebraic integer. Then after a brief review of cyclotomic class field theory, the author proves what he calls the main theorem of complex multiplication, which says that an automorphism of the torsion subgroup is essentially analytic multiplication by an idele of K. This theorem allows one to define a Grossencharacter associated to an elliptic curve with complex multiplication. For such a curve one can then define an L-series and show that it can be expressed as a Hecke L-series with Grossencharacter. ... Read more

Book Description

From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role. The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for the reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996. ... Read more

Book Description

First published in 1952, this book has proven a valuable introduction for generations of students. It provides a clear and systematic development of projective geometry, building on concepts from linear algebra. ... Read more

Book Description

Jean Leray is one of the great French mathematicians of this century. His life's work divides into 3 major areas, reflected in these 3 volumes. Vol. 1, to which an Introduction in English has been contributed by A. Borel, covers Leray's seminal work in algebraic topology, where he created sheaf theory and discovered the spectral sequences. Vol. 2, with an introduction by P. Lax, covers fluid mechanics and PDE: Leray demonstrated the existence of the infinite-time extension of weak solutions of the Navier-Stokes equations; 60 years later this profound work has retained all its impact. Vol. 3, on the theory of several complex variables, has a long introduction by G. Henkin. Leray's work on the ramified Cauchy problem will stand for centuries alongside the Cauchy-Kovalevska theorem for the unramified case. ... Read more

Book Description

The second volume of Shafarevich's introductory book on algebraic varieties and complex manifolds. As with Volume 1, the author has revised the text and added new material, e.g. as a section on real algebraic curves. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum, making the book accessible to non-specialists. It can be read independently of Volume 1 and is suitable for beginning graduate students in mathematics as well as those in theoretical physics. ... Read more

Reviews (1)

After Hartshorne!!!
This book is very good for the secondary course after learning with Harshorne's Algebraic geometry. ... Read more

Book Description

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.

This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory. ... Read more

Reviews (6)

A masterpiece in wavelets
This is a very well written book on the theory of wavelets and its applications. The presentation is self-contained and may serve as an introduction for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts. This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor. It can be certainly used as a textbook for graduate students as well as as a reference book for the specialists and researchers. A very nice feature of the book are the tutorials at the beginning of each chapter and some sections, which serve as summaries of main ideas and guides through the background ideas and motivation. At the end of every chapter there are plenty of excercises of various level of difficulty. Overall, this is a great book and I recommend it highly to a motivated reader.

Distinguished link between math and computer science
This is an important mathematical reference written in excellent style. Wavelets have found applications in many areas of engineering and CS. The authors provide a detailed, rich and entertaining tour through this relatively young but important field for both math and CS/Eng. Connections are, e.g., made between advanced CS virtual-reality applications such as audio-systems processing, future applications such as quantum computing, and advanced math in functional analysis and operator theory.

A fresh look at wavelets
The Book by Bratteli and Jorgensen is a superb book on wavelet's theory. It is very well written and has new and a fresh point of view on the subject. Although there are several good books on wavelets, the book by Bratteli and Jorgensen covers an important niche that has not been covered before. In particular

1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.

2- The writing of the book is very appealing: every chapter starts by a tutorial that gives motivation as well as intuition. It is then followed by a very clean mathematical development of the subject, together with many examples, figures, and applications from physics and engineering. A set of nice problems is provided at the end of each chapter. Thus this book can be used as a graduate textbook or for mathematical seminars in mathematics departments.

3- This book can even be used by experts in wavelet theory for learning about recent developments and new perspectives from operator theory and functional analysis.

I highly recommend this book.

an intriguing new wavelet book
This is a book about an important topic in applied mathematics by two authors with excellent credentials in both pure and applied areas. The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet index theorem, quantum computing, the ubiquitous C*-algebras O_n and, of course, spectral theory. The graphics are meticulously done.

I look forward to learning a lot from it.

An Intriguiging New Book
This is a book about an important topic in applied mathematics by two authors with excellent
credentials in both pure and applied areas.

The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet "index theorem", quantum computing, the ubiquitous Cuntz C*-algebras and, of course, spectral theory. The graphics are excellent. I look forward to learning a lot from it. ... Read more

Book Description

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds. ... Read more

Reviews (7)

A review from a graduate student
If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

algebraic geometry: the real stuff
The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

Work of Art
This is an amazing book with an amazing subject (complex algebraic geometry). Every section presents something interesting and wonderful. I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces). I had had a bad experience with alg geom before this book. Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists. There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices). If you read the book carefully you will get a lot out of it.

Absolutely indispensable
This