A First Course in Modular Forms (Graduate Texts in Mathematics)

Book Description

This book introduces the theory of modular forms with an eye toward the Modularity Theorem. All rational elliptic curves arise from modular forms. The topics covered include: elliptic curves as complex tori and as algebraic curves, modular curves as Riemann surfaces and as algebraic curves, Hecke operators and Atkin-Lehner theory, Hecke eigenforms and their arithmetic properties, the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms, elliptic and modular curves modulo p and the Eichler-Shimura Relation, the Galois representations associated to elliptic curves and to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.

Mirror Symmetry (Clay Mathematics Monographs, V. 1)

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Detailed overview of the subject

Mirror Symmetry (Clay Mathematics Monographs, V. 1)

Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover

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Mirror Symmetry and Algebraic Geometry (Mathematical Surveys & Monographs)
String Theory and M-Theory: A Modern Introduction
Enumerative Geometry and String Theory
Compact Manifolds with Special Holonomy (Oxford Mathematical Monographs)
Calabi-Yau Manifolds and Related Geometries

ASIN: 0821829556

Book Description

This thorough and detailed exposition is the result of an intensive month-long course sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives. The material will be particularly useful for those wishing to advance their understanding by exploring mirror symmetry at the interface of mathematics and physics.

This one-of-a-kind volume offers the first comprehensive exposition on this increasingly active area of study. It is carefully written by leading experts who explain the main concepts without assuming too much prerequisite knowledge. The book is an excellent resource for graduate students and research mathematicians interested in mathematical and theoretical physics.

Customer Reviews:

Detailed overview of the subject.......2005-05-16

Mirror symmetry has become an established branch of mathematics and mathematical physics, and research in the subject has resulted in brilliant developments. This sizable book contains essentially some (polished) lecture notes of a seminar series in mirror symmetry that was given in the spring of 2000. This reviewer only studied Part 5 of the book, entitled "Advanced Topics" and so only that part will be reviewed here. In addition, space constraints then dictate only a small portion of this part can be reviewed. Needless to say, any reader who intends to tackle this book will need a substantial background in modern mathematics and advanced physics, and a sizable commitment in time. The time spent is well worth it though, as both the mathematics and physics behind mirror symmetry has to rank as one of the most fascinating research topics in the last two decades.

In the chapter entitled "Topological Strings" the authors consider the functional integration of worldsheet geometries. This project involves essentially the integration over the complex structures of Riemann surfaces. Referring to this procedure as "quantum gravity", they do not address it in-depth, but instead focus on the coupling of topological sigma models to worldsheet gravity, which is called `topological string theory' in the literature. The authors first consider the case where the target is a Kahler manifold whose first Chern class is zero, since for this case the quantum cohomology ring is less easy to obtain, i.e. it can obtain contributions from holomorphic maps of any degree. Even for the case where there is no coupling to gravity, the degree 0 contribution is related to the classical intersection number. The contributions from higher degree result in the deformation of the classical cohomology ring into the quantum cohomology ring. The authors then ask whether there are any other correlators that will give nontrivial (non-zero) invariants in genus 0. Posing this question leads to the WDVV equation and the genus 0 topological string partition function. The n-point correlation functions of topological strings can then be defined as the nth partial derivatives of this function. For higher genus cases, the correlators are all zero, but the authors show the connection between the higher genus partition function and holomorphic anomalies. The case of three-dimensional Calabi-Yau manifolds is special, if one concentrates on the integration over the complex structures of the worldsheet. When the complex dimension of this moduli space is 3(g-1) then there are isolated points where holomorphic maps exist. Defining a topological string theory for Calabi-Yau threefolds is straightforward, as the author shows, and proceeds analogously to the case of topological field theory. A measure is defined on the moduli space of Riemann surfaces of genus g that cancels the axial charge anomaly. A genus g (>1) topological string amplitude, which is a section of a bundle over the moduli space of Calabi-Yau manifolds, is then obtained from this procedure. Modulo the presence of holomorphic anomalies, the authors show that the definition of topological string amplitudes is consistent with the topological symmetry. The origin of these holomorphic anomalies is discussed in fair detail by the authors, having their origin in the boundaries of the moduli space.

The rigorous mathematical formulation of mirror symmetry is of course of great interest to mathematicians. Because of its origin in string theory and quantum field theory, mirror symmetry has not yet received this kind of rigor. Chapters 37 and 38 of this book discuss some of the approaches that attempt to put mirror symmetry on a more rigorous foundation. One of these involves the use of `derived categories,' an approach that was recommended by the mathematician Maxim Kontsevich. The discussion in these chapters takes place in the context of D-branes, and Kontsevich conjectures that mirror symmetry is the equivalence of two categories: the derived category of coherent sheaves, and the category of Lagrangian submanifolds with flat U(1) connections. Specifically the equivalence entails the equivalence between the bounded derived category of coherent sheaves or `B-cycles' and the category of A-cycles with compositions defined in terms of holomorphic maps from disks. This latter category is derived from the Fukaya A-infinity category, as is shown by the authors. They discuss in detail this category, being essentially a generalization of a differential, graded algebra, especially how to obtain the compositions. In chapter 37, the authors give an explicit example of the equivalence of these categories for the case of the elliptic curve. The elliptic curve is interesting in this regard in that it is its own mirror, i.e. the complex parameter is mapped to the complexified Kahler parameter by the mirror map.

The derived category has sometimes been a stumbling block to those who want to understand the Kontsevich conjecture. The authors do not attempt to give the reader the needed insight into this kind of category, but merely take it to be a collection of all holomorphic bundles and coherent sheaves. Sheaves in this category can be subtracted from each other using a map between them. Physically, this subtraction corresponds to the annihilation of branes and anti-branes via a tachyon. Derived categories though are straightforward to think about if one views them from the standpoint of algebraic topology. Derived categories are rich enough to include notions of localization and triangulated objects (i.e. "complexes") and maps (i.e. morphisms) between these objects. This is a kind of "homology" but what is of main interest are homotopies between the morphisms. The class of homotopic morphisms between two complexes forms an abelian group and one can then obtain a category consisting of complexes as objects and classes of homotopic morphisms as morphisms. A cohomology functor can then be defined on this category, along with graded objects and differentials between them. The homotopic category can be given a "triangulation" and morphisms in this category that give rise to isomorphisms in cohomology are given special status, called `quasimorphisms.' The localization of this category with respect to quasimorphisms is called a derived category.

Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics)

Average customer rating:

Not for the beginner
Good book of reference
very good, but should be read slowly
Superb
The standard text

Commutative Algebra: with a View Toward Algebraic Geometry (Graduate Texts in Mathematics)
David Eisenbud
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Algebraic Geometry (Graduate Texts in Mathematics)
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ASIN: 0387942696

Book Description

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

Customer Reviews:

Not for the beginner.......2007-02-27

Well, the strength of this book lies in where it takes you. There is so much material here that when finished, you'll be prepared for a lot. Personally I think it is too wordy (my preferance is Atiyah & MacDonald) and the typesetting overall isn't all that impressive, so read up or consult other texts before/during your first encounter. M.Reids book is a better place to start.

Good book of reference.......2006-03-18

I purchased this as a book of reference. When I want to know something about Commutative Algebra (while reading Hartshorne's Algebraic Geometrry), I like a standard book of reference. But it seems a good book to learn commutative algebra aswell.

very good, but should be read slowly.......2004-09-25

Some proofs are somewhat abstract to the beginner. Although you are forced to check them on the paper, I think it is very good for the study. Also, you need a professor to instruct you, because in math, any language could only express the part of the oringins. Anyway, algebraic geometry is the course that you have to have a good professor to help you, otherwise stop study this field. In one word, it is a very very good book, so read it slowly!!!!!!

Superb.......2001-09-04

If one is interested in taking on a thorough study of algebraic geometry, this book is a perfect starting point. The writing is excellent, and the student will find many exercises that illustrate and extend the results in each chapter. Readers are expected to have an undergraduate background in algebra, and maybe some analysis and elementary notions from differential geometry. Space does not permit a thorough review here, so just a brief summary of the places where the author has done an exceptional job of explaining or motivating a particular concept:

(1) The history of commutative algebra and its connection with algebraic geometry, for example the origin of the concept of an "ideal" of a ring as generalizing unique factorization.

(2) The discussion of the concept of localization, especially its origins in geometry. A zero dimensional ring (collection of "points") is a ring whose primes are all maximal, as expected.

(3) The theory of prime decomposition as a generalization of unique prime factorization. Primary decomposition is given a nice geometric interpretation in the book.

(4) Five different proofs of the Nullstellensatz discussed, giving the reader good insight on this important result.

(5) The geometric interpretation of an associated graded ring corresponding to the exceptional set in the blowup algebra.

(6) The notion of flatness of a module as a continuity of fibers and a test for this using the Tor functor.

(7) The characterization of Hensel's lemma as a version of Newton's method for solving equations. The geometric interpretation of the completion as representing the properties of a variety in neighborhoods smaller than Zariski open neighborhoods.

(8) The characterization of dimension using the Hilbert polynomial.

(9) The fiber dimension and the proof of its upper semicontinuity.

(10) The discussion of Grobner bases and flat families. Nice examples are given of a flat family connecting a finite set of ideals to their initial ideals.

(11) Computer algebra projects for the reader using the software packages CoCoA and Macaulay.

(12) The theory of differentials in algebraic geometry as a generalization of what is done in differential geometry.

(13) The discussion of how to construct complexes using tensor products and mapping cones in order to study the Koszul complex.

(14) The connection of the Koszul complex to the cotangent bundle of projective space.

(15) The geometric interpretation of the Cohen-Macauley property as a map to a regular variety.

The standard text.......2000-07-28

This is often referred to as the standard text on commutative algebra.

It is an exceptionally good book on a subject that is normally difficult to get a handle on. Eisenbud's readable book gives intuitive and motivated proofs of even very technical results in commutative algebra, often illustrated with instructive examples, such as the useful figures illustrating embedded primes. A very nice feature is that he gives proofs to all the results in commutative algebra used by Robin Hartshorne's popular "Algebraic Geometry," making them a nice pair of books to read together.

I found this to be useful as a reference as well as a text. Most sections are fairly self-contained and many important topics are included in depth. I almost always find that it is the best place to learn any of the material covered.

Book Description

As leading experts in geometric algebra, Chris Doran and Anthony Lasenby have led many new developments in the field over the last ten years. This book provides an introduction to the subject, covering applications such as black hole physics and quantum computing. Suitable as a textbook for graduate courses on the physical applications of geometric algebra, the volume is also a valuable reference for researchers working in the fields of relativity and quantum theory.

Customer Reviews:

makes your head buzz..........2007-08-04

I'm reading this book somewhat in parallel with Hestenes' New Foundations for Classical Mechanics. Both are fantastic books (Hestenes' predates this one), and in some parts they are complementary, while of course they overlap in the foundations and many special topics. What is so fascinating about Geometric Algebra and Calculus? I think it's mainly the recognition that many seemingly complicated theorems of mathematical physics really become much clearer - in a sense of getting a guts feeling about the geometry. The method opens a way to look at the same thing from totally different angles: If one can't imagine something based on geometric arguments, one can take the presented formalism and translate it back into geometry, and suddenly things become clear.
Is the book (or that by Hestenes) basic and easy to understand or are they difficult? Certainly they require some work by the reader. To follow the entire book, one really can't do without learning to master the formalism of geometric algebra, which is simple, yet sometimes bizarre. I suspect though that it is only bizarre to the one who "knows it all" already: The student or scientist who has grown familiar with vector spaces, matrix notation and wiggling around with tensor notation, needs to go through the same exercises as the bloody beginner to whom even the idea of a vector may not be clear. In fact, the beginner could be at a real advantage to not being poisoned by vector calculus. For example, take the very basic notation for a geometric product of two multi-vectors: ab = a.b + a^b (the sum of inner and outer product). What's so confusing about it? Nothing, really, after one really understands what "+" here means. But it happens often enough that one only thinks about this product in terms of the right hand side of the equation, because those are totally familiar for anyone who took basic linear algebra, and then ends up making simple things complicated again. I must say that it was like loosing shadows from the eyes to see how the formulations in this book and Hestenes' work explain so well why it is that the quantum mechanical psi function needs to be complex, or better yet what really the i means in physics, and how the entire set of Maxwell equations (all 4 of them) are one simple continuity equation. That's the kind of thing that makes your head buzz. I'm not done with these books, but I have a clear feeling that in the end I will have an entry point to understand QM and parts of general relativity not just formally (especially QM) but really develop a guts feeling for it.
One thing that I'm still a bit missing in any of the books related to geometric algebra is classical continuum mechanics. This may be so because many of the authors are immersed in fields related to cosmology. In this book, one can find a tiny little bit also about elasticity (linear and nonlinear). However, I keep wondering what it would be like to reformulate the entire underlying theory of continuum mechanics (about deforming solids, elastic or viscoelastic or plastic, about fluid flow, about polarized materials, biological active materials, etc). Could something new be learned? I bet it could!

Provides a very interesting point of view.......2007-02-22

Provides a very interesting point of view, absolutely necessary for grasping the bolts and plumbing of modern physics.

The material covered was not present in other texts that I had a look at so this book serves as a good corner stone to build advanced undergraduate and graduate courses on.

A powerful mathematical language for physics and engineering.......2004-08-01

This is a well-written book on a very interesting and important subject: geometric algebra (GA) is a powerful and elegant mathematical language -- based on the works of Hamilton, Grassmann and Clifford -- that is especially well-suited for spacetime physics and several fields of engineering.

The authors adopt David Hestenes' viewpoint of a graded GA as a unified mathematical language that is coordinate-free, thereby stressing the fundamental role of geometric invariants in physics.

In fact, the elementary vector analysis -- which pervades almost all undergraduate (and even) graduate approaches to electrodynamics -- finds its roots in the misguided Gibbsian approach: Gibbs advocated abandoning Hamilton's quaternions and just work with scalar and cross products of vectors. However, the cross product has a major flaw: it only exists in three (or seven) dimensions -- if we require that (i) it should have just two factors, (ii) to be orthogonal to the factors, and (iii) to have length equal to the corresponding parallelogram.

Electrodynamics and relativistic physics, particularly, are elegantly presented through GA and otherwise cumbersome calculations may be circumvented in a simple and insightful way.

Mainstream physics and engineering cannot overlook GA anymore.

Compared to what ?.......2004-01-30

This is truly a great book for any one who is interested in not just physics, but physical reality. Although the ideas expressed therein have a long history and are by no means as uniquely those of its authors as were Albert Einstein's in his day, I believe that they will have comparable lasting value. Moreover the synthesis presented in this book, which builds pre-eminently on the work of Hestenes, is absolutely superb. Interested readers need not take my word for these claims, but are invited to prove it to themselves.

Although the above should be a sufficient review, my experience nevertheless indicates that it is a good idea to warn potentially enthusiastic readers against several common semantic misconceptions, lest they jump to conclusions which prevent them from ever taking that vital first step. Thus let it be clearly understood that Geometric Algebra is NOT:
(1) A replacement for linear/matrix/tensor algebra (on the contrary, it is a very nice complement to these formalisms).
(2) Identical, or even very close, to Emil Artin's earlier excellent book on bilinear forms with the title "Geometric Algebra".
(3) Another name for the enormous field "algebraic geometry" (it is indeed appropriate that the word stemming from "geometry" comes first in "geometric algebra").
(4) Just another reformulation of complex / quaternion / octonian analysis; for it connects all these purely algebraic objects, and many generalizations thereof, to Felix Klein's Erlangen Programme and Sophus Lie's theory of continuous groups.
(5) The ultimate theory of everything (although it probably will eventually be found to have something to do with it).

Geometric algebra IS a practical and natural (canonical) tool for formulating physical and mathematical problems in homogeneous spaces in a fully covariant fashion. But more importantly, you do not need to understand all those words in order to benefit from it, and this book is an excellent place for physicists of all stripes to start.

Articulate Path to the Future.......2003-07-19

The quality and importance of this book could hardly be overstated. Geometric algebra might casually be considered the "correct" generalization of linear algebra. By considering, for a start, directed line segments, the linear algebra courses presently taught in some high schools and all universities achieve miracles. Although viewed by a few of the slower students as merely unpleasant bookkeeping systems, linear algebra derives its power from allowing algebraic manipulation of sophisticated aggregate objects, namely vectors. The benefits are not just computational, but stem more importantly from a more powerful and more unified, although slightly more abstract point of view than a student had before studying. Geometric algebra is all that and much more. By extending consideration from directed line segments to the inclusion of direct plane segments, directed elements of three space, etc., an extremely flexible and elegant mathematical tool arises. It allows a deeper, quicker, and more concise treatment of essentially all of modern differential geometry. Its applications throughout physics are at once simplifications of ordinary matrix treatments and occasions to allow much greater insight.

Geometric algebra is a great theory, one of highest importance. It will, undoubtedly, find a dominant place in our mathematics curriculum at the highest speed allowed by our educational systems (the highest speed being actually quite slow). This book is an especially good place to begin study. It starts from the most elementary principles, and exposes the material with very thoughtful, clear presentation. The economy and elegance of the geometric algebra itself allows this one substantial but not enormous book to reveal great insights into many branches of study, from differential geometry and its applications to gravity theory to quantum mechanics and classical mechanics.

Book Description

This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Customer Reviews:

Excellent.......2001-12-22

Who would have known that the equation discovered by P.A.M. Dirac in the 1920's would have the enormous appllications to mathematics that it currently has. This book is an excellent overview of these applications, written by two individuals who are responsible for the development of many of these. Dirac's theory of course had its origins in physics, and physicists, particularly those working in high energy physics, will find this book interesting and helpful.

The authors give a brief introduction and then move on to the representation theory of Clifford algebras and spin groups in chapter 1. The reader can see the origin of Clifford algebras and an introduction to the Pin and Spin groups. Clifford algebras are classified as matrix algebras over the real or complex numbers, and the quaternions. It is the representation theory of Clifford algebras however that has resulted in the impressive results outlined in the book Noting that the tensor product of Clifford algebras is not necessarily a Clifford algebra, the authors introduce a Z(2)-grading on a Clifford algebra, which results in a multiplicative structure in the representations of Clifford algebras. The Lie algebras of the Pin and Spin groups are discussed along with applications to geometry and Lie groups. By far the most interesting discussion though is on K-theory, which allows one to define a ring structure on vector bundles. Distinguishing a base point in the base space, relative K-groups are defined, and shown to be equal for the base space and its i-fold suspension. Bott periodicity results are stated but their proof is delayed until chapter 3. A detailed discussion is given of the Atiyah-Bott-Shapiro isomorphism and KR-theory.

The connection between spin and differential geometry is discussed in chapter 2. The first few sections is a review of standard results in the spin structure of vector bundles, such as Stiefel-Whitney classes and spin cobordism. For Riemannian vector bundles, each fiber has a quadratic form that gives rise to a Clifford algebra on the fiber. The question as to when a vector bundle over the Riemannian base space can be found that has fibers each an irreducible module over this Clifford algebra leads to a consideration of spin manifolds and spin cobordism, when the total space is chosen to be the tangent bundle. The Dirac operator acting on a bundle over this Clifford bundle allows the construction of all the standard elliptic operators such as the signature, Atiyah-Singer, and the Euler characteristic. The authors discuss these constructions in detail along with the notion of of Cl(k)-linear operators.

The Dirac operator can be viewed in Euclidean space as the square root of a Laplace operator, but over general manifolds it is the Laplacian with a correction term dependent on the curvature and Clifford multiplication. The Bochner vanishing theorems are discussed in great detail, along with the results on the existence of exotic spheres.

An entire chapter is spent on index theorems, wherein the authors present the results in terms of the approach used by Atiyah and Singer, instead of the heat kernel methods of Gilkey and Patodi. Physicists might prefer the later approach, due to its connections with applications, but the abstract K-theory approach undertaken by the authors is elegant and their presentation is excellent. The role of physics in index theorems is a fascinating one though, especially the use of supersymmetry to simplify the proofs of some of the results. The authors do not discuss this approach, but point out, interestingly, that it does not work when one is dealing with torsion elements in K-theory. These cannot be detected using cohomology nor can the modulo-two invariants appearing in the index theorems be computed from local densities.

The last chapter is a long one and discusses applications in differential topology and geometry, emphasizing index thoerems and Riemannian manifolds of positive scalar curvature. The authors outline just when the indexes are integers (the integrality theorems) and use spin geometry to discuss the immersion problem for manifolds and the vector field problem. Exotic n-spheres again make their appearance, wherein it is shown that some of these have very few symmetries and are very asymmetric objects. A short introduction to elliptic genera is given. Interestingly, C*-algebras are briefly mentioned as tools to decide whether for every compact spin manifold with positive scalar curvature all higher A-genera must be zero. Spin-c manifolds are not treated, the authors instead concentrating their attention to Kahlerian geometry. In this context the Clifford algebra multiplication has a beautiful relationship with the complex structure. A brief discussion is given of the pure spinors of Cartan and twistor spaces. The theory of holonomy and calibrations, the later due to one of the authors, is discussed in great detail. The discussion begins in the consideration of when universal covering spaces are not Riemannian manifolds and their holonomy groups have been classified. The idea of a calibration arises from the consideration of submanifolds that are homologically volume-minimizing. These become calibrations when the integrals of p-forms on them are the volumes, and these p-forms have vanishing differentials on oriented tangent p-planes on the manifold. The authors give an interesting discussion of the relation between spinors and calibrations.

Essential for grad students in geometry/topology.......1998-12-23

As a graduate student in mathematics I survived on this encyclopedic work. Anyone interested in differential geometry or differential topology will eventually need something in this book.

Prerequisites are graduate-level algebra and analysis, and some topology and differential geometry. He introduces the subject of pseudodifferential operators and Sobolev spaces, but it's easy to get lost in that part unless you first read Shubin's book "Pseudodifferential operators and Spectral theory". Also, the quick shuffling of Lie group information can be disheartening if you're not used to it. Harvey's book "Spinors and Calibrations" is a more elementary book if this is the case.

Book Description

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature—from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on a fascinating tour of the many ramifications of the Fibonacci numbers. The authors begin with a brief history of their distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few. Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

Customer Reviews:

A complicated subject presented in a very uncomplicated manner........2007-08-30

The book provides much of the available information on the Fibonacci numbers. It starts with the life of Leonardo da Pisa, the man who first introduced the numbers to the world almost a thousand years ago. It describes the actual sequence, then demonstrates the connection that the numbers have to the natural, as well as to the world of the visual arts and of music. Even the stock market is not immune of the influence of the Fibonacci sequence.
What particularly impressed me about this book is the clarity with which the authors present the subject. Whether you are a mathematician or simply have an inquisitive mind, you will always know the exact meaning of the subject under discussion. In fact, you can skip the (sometimes) long mathematical formulae and still never lose track of the narrative.
A wonderful book that makes one ponder on the origin and significance of the created world. A must for mathematicians, scientists and generally educated individuals. A must also for those who believe that our universe and all its contents are only the product of a series of coincidences. These people may change their minds after becoming familiar with the Fibonacci numbers.

Encompassing and Interesting.......2007-08-23

The book contains many interesting and unpredictable properties of the Fibonacci numbers. I learned a lot of new things about them.

Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs)

Average customer rating:

Self contained introduction to techniques of classifying manifolds.
A very good book.

Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs)
Shigeyuki Morita
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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Similar Items:

The Geometry of Physics: An Introduction, Second Edition
A Geometric Approach to Differential Forms
Geometry, Topology and Physics, Second Edition (Graduate Student Series in Physics)
Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics)
Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)

ASIN: 0821810456

Book Description

Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.

The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.

The book can serve as a textbook for undergraduate students and for graduate students in geometry.

Customer Reviews:

Self contained introduction to techniques of classifying manifolds........2007-01-10

This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).

For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.

Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.

Modern approaches use differential forms to represent homology and cohomoly groups.

The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.

Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.

There are plenty of exercises to boot.

A very good book........2005-03-28

This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition. I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book.

The Geometry of Four-Manifolds (Oxford Mathematical Monographs)

Average customer rating:

An excellent summary of Donaldson theory

The Geometry of Four-Manifolds (Oxford Mathematical Monographs)
S. K. Donaldson , and P. B. Kronheimer
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Paperback

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ASIN: 0198502699

Book Description

This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result have had far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.

Customer Reviews:

An excellent summary of Donaldson theory.......2000-06-16

This book brings together the brilliant work Donaldson did at Oxford during the early 1980s. The unique properties of 4-manifolds are clearly and concisely written out with concentration on explaining field theories like Yang-Mills and gauge theory with a truly firm mathematical foundation, presented in a book for the first time. A great companion for any researcher in the field of geometry and topology, or even loop quantum gravity!

Introduction to Coding Theory (Graduate Texts in Mathematics)

Average customer rating:

Excellent book from mathematical standpoint

Introduction to Coding Theory (Graduate Texts in Mathematics)
J.H. van Lint
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
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ASIN: 3540641335

Book Description

From the reviews: "The 2nd (slightly enlarged) edition of the van Lint's book is a short, concise, mathematically rigorous introduction to the subject. Basic notions and ideas are clearly presented from the mathematician's point of view and illustrated on various special classes of codes...This nice book is a must for every mathematician wishing to introduce himself to the algebraic theory of coding." European Mathematical Society Newsletter, 1993 "Despite the existence of so many other books on coding theory, this present volume will continue to hold its place as one of the standard texts...." The Mathematical Gazette, 1993

Customer Reviews:

Excellent book from mathematical standpoint.......2005-02-20

Very good intro textbook. It gives short, detailed preps to various coding areas (linear, cyclic, convolutional). The biggest advantage this book has is that it does not throw at You tonnes of unnecessary info (like many other thick books do). That is, it assumes reader has some basic understanding of algebra and probability theory. Let's say, it gives good theoretical presentation such that the reader gets good theoretical understanding, it is not example-based.

Mandatory Package College Algebra with Trigonometry with Smart CD (Windows)

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Mandatory Package College Algebra with Trigonometry with Smart CD (Windows)
A college student from MCPV

Mandatory Package College Algebra with Trigonometry with Smart CD (Windows)
Raymond A. Barnett , Michael R. Ziegler , and Karl E. Byleen
Manufacturer: McGraw-Hill Science/Engineering/Math
ProductGroup: Book
Binding: Hardcover

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ASIN: 0072412186

Book Description

The Barnett, Ziegler, Byleen College Algebra/Precalculus series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. College Algebra with Trigonometry, 7/e, introduces a right triangle approach to trigonometry and can be used in one or two semester college algebra with trigonometry or precalculus courses. The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, the students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept that is being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD tutorial reinforces important concepts, and provides students with extra practice problems.

Customer Reviews:

Mandatory Package College Algebra with Trigonometry with Smart CD (Windows).......2006-09-22

This book was confusing. I found descrepancies. It really frustrated me. If you are an online student, good luck with this book. You will definately need tutoring.

A college student from MCPV.......2000-08-10

This book may be expensive but is really helpful, not only in class but also as a reference or as a study guide. The language used in this book is simple and the graphics and images easy to understand. For me trigonometry was always a pain and I got an A in the class, so I decided to write a review for future students.

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