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This book contains an introduction to the recently developedspectral theory of associative rings and Abelian categories, and itsapplications to the study of irreducible representations of classes ofalgebras which play an important part in modern mathematical physics.
Audience: A self-contained volume for researchers and graduatestudents interested in new geometric ideas in algebra, and in thespectral theory of noncommutative rings, currently invadingmathematical physics. Valuable reading for mathematicians working onrepresentation theory, quantum groups and related topics,noncommutative algebra, algebraic geometry, and algebraic K-theory.
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These lecture notes contain a guided tour to the Novikov Conjecture and related conjectures due to Baum-Connes, Borel and Farrell-Jones. They begin with basics about higher signatures, Whitehead torsion and the s-Cobordism Theorem. Then an introduction to surgery theory and a version of the assembly map is presented. Using the solution of the Novikov conjecture for special groups some applications to the classification of low dimensional manifolds are given. Finally, the most recent developments concerning these conjectures are surveyed, including a detailed status report.
The prerequisites consist of a solid knowledge of the basics about manifolds, vector bundles, (co-) homology and characteristic classes.

Customer Reviews (1)

An effective overview
In its simplest form, the Novikov conjecture asserts that if there is a map f from a closed smooth manifold M and a classifying space of a group, and if g is a homotopy equivalence from a closed smooth manifold N to M, then the `higher signatures' of (M, f) and (N, fg) agree. The goal of this book is to introduce the reader to the precise notion of `higher signature' and to discuss various concepts and tools used in attempted resolutions of the conjecture. Also discussed in some details are conjectures that are related to the Novikov conjecture. Readers will needto have a strong background in algebraic and differential topology in order to appreciate the content of the book, but the authors develop some of the needed material in it, such as h- and s-cobordism, simple homotopy, surgery theory, and the classification problem for manifolds via characteristic classes. Without too many exceptions the authors motivate the concepts exceedingly well, especially in chapter 12 where they give one of the best explanations in print for the surgery obstruction groups.

When reading the book it becomes apparent that the Novikov conjecture has many guises, and attempts to resolve it have involved some quite esoteric constructions. The main strategy used in its resolution involves a generalization of the Hirzebruch signature, called a `higher signature' and the notion of an `assembly map.' The assembly map, as the name implies, collects all the higher signatures into a single invariant: essentially the image of the Poincare dual of the L-class under the map induced from f. One then constructs a homomorphism (the assembly map) from the Poincare duals of the Pontrjagin classes to a particular Abelian group L(G), such that the value of the assembly map on the image is a homotopy invariant. The Novikov conjecture is the assertion that the assembly map is an isomorphism. Much of the first part of the book discusses how to make these notions meaningful and how to interpret them geometrically via the surgery obstruction groups.

The authors also discuss them in a purely algebraic context, constructing an algebraic notion of bordism in the context of chain complexes and the notions of symmetric and quadratic forms over chain complexes. Algebraic cobordism allows the definition of a symmetric and quadratic algebraic L-group. The nth symmetric algebraic L-group of a ring R with involution is defined as the collection of cobordism classes of n-dimensional symmetric algebraic Poincare complexes, and the quadratic L-group of R, with a similar definition for the quadratic case. From these constructions the reader is introduced to the subject of L-theory, which has been the subject of intense research in the last two decades.

Central to the research into the Novikov conjecture is the category of `spectra', which is usually encountered in any treatment of algebraic topology but is discussed here with examples given in K- and L-theory and the famous Thom spectrum of a stable vector bundle. The discussion of spectra involves the important notion of a `homotopy pushout', which are defined so as to commute with the suspension with the unit circle, and the `homotopy pullback', which commutes with the loop functor. Both homotopy pushouts and pullbacks are homotopy equivalences.

Given a discrete group G, a family of subgroups of G, and an equivariant homology theory with respect to G, after constructing the classifying space of the family of subgroups, the authors want to show that the assembly map induced from the projection of the family to the one-point space is an isomorphism. To understand for which groups this is true, the authors must first define the notion of a G-homology theory. They use the notion of a G-CW-complex that they defined when discussing classifying spaces of families of subgroups to define this homology theory. For a group G and an associative commutative ring Q with unit it consists of a collection of covariant functors from the category of G-CW-pairs to the category of Q-modules indexed by the integers that satisfies the usual properties such as G-homotopy invariance, the existence of a long exact sequences for pairs, and excision. An equivariant homology theory is then a G-homology theory that has a `induction structure', the latter of which is a collection of isomorphisms between the nth G-homology groups and nth homology groups of a group that has a homomorphism into G. The authors then show how to obtain an equivariant homology from a spectrum. Central to their construction is the `orbit category Or(G)' of a group G. The objects of this category are homogeneous G-spaces and the morphisms are G-maps. For a small category C they define a `C-space' to be a functor from C to the set of compactly generated spaces. A `C-spectrum' is a functor from C to the category of spectra. After defining a notion of smash product for a C-space and a C-spectrum, the authors then quote a lemma that illustrates how one can obtain a G-homology theory from an Or(G)-spectrum. In order to obtain an induction structure, the Or(G)-spectrum must be obtained from a spectrum of groupoids. The authors show how to do this and thus obtain an equivariant homology theory. Therefore the K- and L-theory spectra over groupoids that were constructed earlier thus give rise to equivariant homology theories.

These G-homology theories reduce to the K- and L-theory of the group ring when evaluated on a one-point space, and the topological K-theory of the reduced C*-algebra. The Farell-Jones conjecture claims that the assembly maps from the equivariant homology groups for K and L-theory to the one-point homology are isomorphisms. The Baum-Connes conjecture does the same for topological K-theory. If these conjectures were answered positively then they would allow the computation of the K- and L-groups from the K- and L- finite or virtually cyclic subgroups of G. The authors spend two chapters discussing for what groups these conjectures have been found to be true, and also a chapter on how the Novikov conjecture follows from these conjectures. ... Read more

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Arithmetic noncommutative geometry uses ideas and tools from noncommutative geometry to address questions in a new way and to reinterpret results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at Archimedean places of arithmetic surfaces and varieties. Noncommutative geometry can be expected to say something about topics of arithmetic interest because it provides the right framework for which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry. With a foreword written by Yuri Manin and a brief introduction to noncommutative geometry, this book offers a comprehensive account of the cross fertilization between two important areas, noncommutative geometry and number theory. It is suitable for graduate students and researchers interested in these areas. ... Read more

Customer Reviews (1)

Compact and to the point
This book seems to be very helpful for the relatively new and growing subject. Anyways, be cautious that this book is not a hardcover edition. Only softcover edition got out, I believe. See AMS website. ... Read more

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"The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available." W. Kleinert in Z.blatt f. Math., 1992 "The author's enthusiasm for this topic is rarely as evide!nt for the reader as in this book. - A good book, a beautiful book." F. Lorenz in Jber. DMV 1995 "The present work is written in a very careful and masterly fashion. It does not show the pains that it must have caused even an expert like Neukirch. It undoubtedly is liable to become a classic; the more so as recent developments have been taken into account which will not be outdated quickly. Not only must it be missing from the library of no number theorist, but it can simply be recommended to every mathematician who wants to get an idea of modern arithmetic." J. Schoissengeier in Montatshefte Mathematik 1994 ... Read more

Customer Reviews (3)

Must have for AN
This book is great.It takes a different approach than some other texts I have recently read but is an excellent starting point for those interested in AN.

One of the most beautifully written math books
This book is basically all you need to learn modern algebraic number theory. You need to know algebra at a graduate level (Serge Lang's Algebra) and I would recommend first reading an elementary classical Algebraic number theory book like Ian Stewart's Algebraic Number Theory, or Murty and Esmonde's Problem's in Algebraic Number theory.

10 stars if I could.
After having no fun with Lang's text "Algebraic Number Theory" I began seking out something more complete and which was full of quality exposition.As a result of Amazon's approach to marketing towards members, I was recommended this book and decided quickly that I must have it.This book is marvelously well written, examples are kept to an un-overwhelming minimum, the problems are not trivial (at least to me) and in fact I feel this is the kind of book on par with, say, Paulo Ribenboim's "Classical Theory of Algebraic Numbers" since these are both the type of book you would want to take with you on a long trip or as Paulo says, "while stranded on a desert island".This book is by no means intended for those who are not fluent in both Number Theory as well as Algebra, both at the graduate level and obviously for those who are Mahematically gifted.I highly recommend this book to graduate students interested in Algebraic number theory as well as those needing a splendid reference. ... Read more

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Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.

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The study of the zeroes of polynomials, which for one variable is essentially algebraic, becomes a geometric theory for several variables. In this book, Fischer looks at the classic entry point to the subject: plane algebraic curves. Here one quickly sees the mix of algebra and geometry, as well as analysis and topology, that is typical of complex algebraic geometry, but without the need for advanced techniques from commutative algebra or the abstract machinery of sheaves and schemes.In the first half of this book, Fischer introduces some elementary geometrical aspects, such as tangents, singularities, inflection points, and so on. The main technical tool is the concept of intersection multiplicity and Bézout's theorem. This part culminates in the beautiful Plücker formulas, which relate the various invariants introduced earlier. The second part of the book is essentially a detailed outline of modern methods of local analytic geometry in the context of complex curves. This provides the stronger tools needed for a good understanding of duality and an efficient means of computing intersection multiplicities introduced earlier. Thus, we meet rings of power series, germs of curves, and formal parametrizations. Finally, through the patching of the local information, a Riemann surface is associated to an algebraic curve, thus linking the algebra and the analysis. ... Read more

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The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly, Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises offering useful results in complex algebraic geometry. Also available: Volume I 0-521-80260-1 Hardback $60.00 C ... Read more

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Algebraic geometry plays an important role in several branches of science and technology. This is the last of three volumes by Kenji Ueno algebraic geometry. This, in together with Algebraic Geometry 1 and Algebraic Geometry 2, makes an excellent textbook for a course in algebraic geometry.

In this volume, the author goes beyond introductory notions and presents the theory of schemes and sheaves with the goal of studying the properties necessary for the full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. Ueno also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. ... Read more

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This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

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Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introduction to Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristics and Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. ... Read more

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Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other subdisciplines. This book provides a comprehensive overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers, and useful to people working in the applied fields.

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Few people have proved more influential in the field of differential and algebraic geometry, and in showing how this links with mathematical physics, than Nigel Hitchin. Oxford University's Savilian Professor of Geometry has made fundamental contributions in areas as diverse as: spin geometry, instanton and monopole equations, twistor theory, symplectic geometry of moduli spaces, integrables systems, Higgs bundles, Einstein metrics, hyperkähler geometry, Frobenius manifolds, Painlevé equations, special Lagrangian geometry and mirror symmetry, theory of grebes, and many more.

He was previously Rouse Ball Professor of Mathematics at Cambridge University, as well as Professor of Mathematics at the University of Warwick, is a Fellow of the Royal Society and has been the President of the London Mathematical Society.

The chapters in this fascinating volume, written by some of the greats in their fields (including four Fields Medalists), show how Hitchin's ideas have impacted on a wide variety of subjects. The book grew out of the Geometry Conference in Honour of Nigel Hitchin, held in Madrid, with some additional contributions, and should be required reading for anyone seeking insights into the overlap between geometry and physics. ... Read more

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An accessible text introducing algebraic geometry and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic geometries from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups and Lie type. The text covers the conjugacy of borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, a thorough treatment of Frobenius maps on affine varieties and algebraic groups, zeta functions, and Lefschetz numbers for varieties over finite fields. Experts in the field will enjoy some of the new proofs. The text uses algebraic groups as the main examples, including worked out examples, instructuve exercises, as well as bibliographical and historical remarks. ... Read more

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Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry – random sets, point processes, random mosaics – and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

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The book is devoted to the properties of conics (plane curves of second degree) that can be formulated and proved using only elementary geometry. Starting with the well-known optical properties of conics, the authors move to less trivial results, both classical and contemporary. In particular, the chapter on projective properties of conics contains a detailed analysis of the polar correspondence, pencils of conics, and the Poncelet theorem. In the chapter on metric properties of conics the authors discuss, in particular, inscribed conics, normals to conics, and the Poncelet theorem for confocal ellipses. The book demonstrates the advantage of purely geometric methods of studying conics. It contains over 50 exercises and problems aimed at advancing geometric intuition of the reader. The book also contains more than 100 carefully prepared figures, which will help the reader to better understand the material presented ... Read more

Customer Reviews (1)

A lovely book !!
There is a definite dearth of modern books dealing with geometrical conics, that isto say using the methods of classical euclidean and projective geometry to derive their properties. In this respect Akopyan's book should be warmly welcomed.

A few other points pertaining to what used to be called Modern Geometry, such as cevians, symmedians, Lemoine and Brocard points, Simson lines,and some of their properties are also presented to new generations of readers.

Much of this stuff used to be taught in this way in the 19th and early 20th century (cfr. C. V. Durell's delightful books), but later fell out of fashion. Fortunately a revival of interest in this classical way of teaching geometry can be perceived these days.

I've only read part of the book so far, but I must admit it is a lovely book.

However I find the book a bit beyond "... the reach of high school students", as the pace is rather brisk. Particularly projective geometry definitely deserves a longer and more detailed introduction.

There is a mistake in the definition of parabola in the last paragraph of page 2 (line 3 from bottom), where "equal" should be substituted for "constant".
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Customer Reviews (9)

If U want generalization out to infinity, this is it for you, in algebraic topology basics.
This former professor, and sometime head of the math department at U of Chicago, is quite a fellow. He is so DEEP that I many times didn't have a clue about some of his books. But this one seems more down at my level of intelligence, even though it is a whirlwind romance so to speak with algebraic topology basics.

Have at it, if you like the whirlwind!

The Title Says it All
I have always believed that the "goodness" of a mathematical textbook is inversely proportional to its length. J. P. May's book "A Concise Course in Algebraic Topology" is a superb demonstration of this. While the book is indeed extremely terse, it forces the reader to thoroughly internalize the concepts before moving on. Also, it presents results in their full generality, making it a helpful reference work.

The opposite of Hatcher
This book is clear, and direct.It tells you want you want to know.

Lucid and elegant, but not for beginners
This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

A Unique and Necessary Book
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory. ... Read more

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In recent years noncommutative geometry has been a richtopic of research with discoveries leading to an increasing number ofapplications in mathematics and theoretical physics.Very little hasappeared in book form since Alain Connes' work in the early 90s todeal with this subject."Elements of Noncommutative Geometry" fillsan important gap in the literature.

Key features of the work include: * unified and comprehensive presentation of core topics and key research results drawing from several branches of mathematics
* rigorous, well-written, nearly self-contained exposition of noncommutative geometry and some of its useful applications to quantum theory
* excellent exposition of introductory material; main topics covered repeatedly in the text at gradually more demanding levels of difficulty
* many applications to diverse fields: index theory, foliations, number theory, particle physics, and fundamental quantum theory
* rich in proofs, examples and exercises
* comprehensive bibliography and index

This text is an introduction to the language and techniques ofnoncommutative geometry at a level suitable for graduate students, andalso provides sufficient detail to be useful to physicists andmathematicians wishing to enter this rapidly growing field.It mayalso serve as a reference text on several topics that are relevant tononcommutative geometry. ... Read more

Customer Reviews (2)

right complement
Pepe and Joe have been involved in NCG research
since the first physical calculations. And
the book happened to be published in a very
adequate time range, so it contains up-to-date
information on all the developments, including
the finding of Connes-Kreimer renormalization
algebra (hopf algebras and butcher groups).

It is a compulsory complement to Connes's book.

right complement
Pepe and Joe have been involved in NCG research
since the first physical calculations. And
the book happened to be published in a very
adequate time range, so it contains up-to-date
information on all the developments, including
the finding of Connes-Kreimer renormalization
algebra (hopf algebras and butcher groups).

It is a compulsory complement to Connes's book. ... Read more