Product Description
Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title. ... Read more

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Progress in low-dimensional topology has been very fastover the last two decades, leading to the solutions of many difficultproblems. One of the consequences of this "acceleration of history" isthat many results have only appeared in professional journals andmonographs. These are hardly accessible to students who have completedonly a basic course in algebraic topology, or even to some researcherswhose immediate area of expertise is not topology.

Among the highlights of this period are Casson’s results on the Rohlininvariant of homotopy 3-spheres, as well as his l-invariant. Thepurpose of this book is to provide a much-needed bridge to thesemodern topics. The book covers some classical topics, such as Heegaardsplittings, Dehn surgery, and invariants of knots and links. Itproceeds through the Kirby calculus and Rohlin’s theorem to Casson’sinvariant and its applications, and gives a brief sketch of links withthe latest developments in low-dimensional topology and gauge theory.

The book will be accessible to graduate students in mathematics andtheoretical physics familiar with some elementary algebraic topology,including the fundamental group, basic homology theory, and Poncaréduality on manifolds. ... Read more

Customer Reviews (2)

great intro to low-dimensional topology
This is a great intro book for someone interested in low-dimensional topology. It is very clear and well-written, assumes only what you can find in an introduction to topology like Hatcher's, and gives an intuitive explanation of many important results which are a prerequisite for working in low-dimensional topology, although, since it is just an intro, it does not include full proofs. I would recommend this book for someone curious to learn some basic facts, and it's a fantastic way for a new topologist to get introduced to the field. For example, it includes a useful section on Kirby calculus which eases the jump to in-depth works like Gompf and Stipsicz.

good book
1. It is a beautiful book. There are many pictures in it. They can help you to understand concepts and proof.
2. The book begins with basic content but end with considerable deep result. Reading this book doesn¡¯t require much prior knowledge of 3-manifolds. You can read it quickly and get much.
3. The author successfully explains the Casson invariant in an elementary and readable style. He gives a short chapter to basic tools of topology. Then he describes Hergarrd splitting, Kirby moves, Dehn surgery, and intersections of 4-manifolds, Rochlin invariant, Arf invariant and representation of Lie group. At last he defines the Casson invariant and calculate some examples.
4. But there is some shortcoming. Some results are not proofed. Something needs more explanation. You can find some content is completely moved from other books. It is not natural.
5. Anymore, I recommend this book to everyone who has interest in topology. If you are no interested in topology, just read it. I bet you will start liking it. ... Read more

Product Description

This is an extended second edition of The Topology of Torus Actions on Symplectic Manifolds published as PM 93 in 1991. The material and references have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains more theorems, more (and better) proofs, more exercises, and many figures. This enlarged version, although it includes all the material treated in the first edition, is indeed a new book.

Product Description
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'. ... Read more

Customer Reviews (5)

A graduate physics student
If you are in a technical field of study and keep seeing words like, manifold, tangent bundle, vector bundle, lie groups ..., and want to know what they are, this book is a must. For anyone studying General Relativity, Quantum Mechanics or Quantum field theory, this book can clear up some of the mathematical confusion of these subjects.

This book covers the mathematics included in most upper level science courses that is not covered in the required math courses. This book is easy enough to understand to give any one in a technical field an edge. This book help me so much, that I was able to go and read more abstract math books.

This book should be required reading for anyone entering graduate level physics. If you are like me and really want to understand physics, or any of the sciences at a higher mathematical level then get this book.

Un-illuminating exercises, full of errata
I find this book to be a very frustrating textbook for my graduate-level differential geometry course.The exercises are un-illuminating verifications of simple problems, and don't really help me understand the concepts.The book is chock full of typos, many (but by no means all) of which are listed in an errata sheet available at [...].Among the most egregious is that in the section on point-set topology, the author defines a closed map as one which maps closed sets to open sets.

The book is not entirely without merit.The actual body of the text provides a reasonable introduction to manifold theory.Despite an extensive background in point-set topology, algebraic topology, and algebra, I still find the subject exceptionally confusing.

Clear and Solid Exposition
This is my favorite book on Differentiable Manifolds. After reading this book the reader will obtain a solid background on the following essential notions: Charts and atlas of a manifold; tangent vectors (as derivations); differential of a smooth function between manifolds; submanifolds and embeddings; quotient spaces; partitions of unity; vector fields; vector bundles; differential forms and de Rham cohomology. And on the road, the reader gets a gentle exposure to Lie groups, Lie algebras; and some basic notion of Category and Functors.

I found the following aspects of the book especially attractive:
> Clear style of writing: The author is the coauthor of the acclaimed "Differential Forms in Algebraic Topology". See the comments for that book. The clarity has not decreased at all.
> Bite-sized sections: The materials contained in each section is approximately equal to that of a 50-minute lecture. This helps readers who plan self-study.
> Right amount of topics: This is not an encyclopedia on manifolds. However, it does contain the ``absolute must'' one should know about manifolds. And it does such a good job in presenting it, the reader will be left with a solid understanding on those essential topics.

I first read this book as a Physics student and had no trouble reading it. I later switched discipline to Mathematics, and I know that this book has helped me appreciate the beauty of Mathematics. I thank the author for writing such an wonderful book.

An introduction it Manifolds
This is an excellent book. I wish that more books on advanced mathematics were written in this style. In contrast to most books on manifolds that tend to be very difficult for beginners to follow, Prof. Tu has made every effort to make this subject understandable to the nonexpert.

Greg Chirikjian
Professor, Mechanical Engineering
Johns Hopkins University

fills a gaping hole

i think there is a jump from ugrad analysis/alg/top etc to early grad school concepts. i didnt know category theory, i only had the flimsiest notion of a manifold, etc etc. and this book fills in that jump wonderfully. it does the right mix of analysis-differential topology-topology so that you can go read a book like bott and tu later (that's what it was designed for).
so im having a good time with it. ... Read more

Product Description
The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology. The book's guiding philosophy is, in the words of Newton, that "in learning the sciences, examples are of more use than precepts". We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue. While we present most of the local aspects of classical differential geometry, the book has a "global and analytical bias". We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincare duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades.Thus, the last part of the book discusses elliptic equations, including elliptic Lp and Holder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators. The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight. ... Read more

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From the reviews of the second edition: "The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. The differential geometrical methods of this theory were developed essentially under the influence of Professor S.-S. Chern's works. The present book is a second edition... It can serve as an introduction to, and a survey of, this theory and is based on the author's lectures held at the University of California and at a summer seminar of the Canadian Mathematical Congress.... The text is illustrated by many examples... The book is warmly recommended to everyone interested in complex differential geometry." #Acta Scientiarum Mathematicarum, 41, 3-4# ... Read more

Product Description
The goal of these lectures is to present an introduction to the geometric topology of the Hilbert cube Q and separable metric manifolds modeled on Q, which are called here Hilbert cube manifolds or Q-manifolds. In the past ten years there has been a great deal of research on Q and Q-manifolds which is scattered throughout several papers in the literature. The author presents here a self-contained treatment of only a few of these results in the hope that it will stimulate further interest in this area. No new material is presented here and no attempt has been made to be complete. For example, the author has omitted the important theorem of Schori-West stating that the hyperspace of closed subsets of $[0,1]$ is homeomorphic to Q. In an appendix (prepared independently by R. D. Anderson, D. W. Curtis, R. Schori and G. Kozlowski) there is a list of problems which are of current interest. This includes problems on Q-manifolds as well as manifolds modeled on various linear spaces. The reader is referred to this for a much broader perspective of the field. In the first four chapters, the basic tools which are needed in all of the remaining chapters are presented. Beyond this there seem to be at least two possible courses of action. The reader who is interested only in the triangulation and classification of Q-manifolds should read straight through (avoiding only Chapter VI). In particular the topological invariance of Whitehead torsion appears in Section 38. The reader who is interested in R. D. Edwards' recent proof that every ANR is a Q-manifold factor should read the first four chapters and then (with the single exception of 26.1) skip over to Chapters XIII and XIV. ... Read more

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Author is well-known and established book author (all Serge Lang books are now published by Springer); Presents a brief introduction to the subject; All manifolds are assumed finite dimensional in order not to frighten some readers; Complete proofs are given; Use of manifolds cuts across disciplines and includes physics, engineering and economics ... Read more

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In 1901, Karl Pearson invented Principal Component Analysis (PCA). Since then, PCA serves as a prototype for many other tools of data analysis, visualization and dimension reduction: Independent Component Analysis (ICA), Multidimensional Scaling (MDS), Nonlinear PCA (NLPCA), Self Organizing Maps (SOM), etc. The book starts with the quote of the classical Pearson definition of PCA and includes reviews of various methods: NLPCA, ICA, MDS, embedding and clustering algorithms, principal manifolds and SOM. New approaches to NLPCA, principal manifolds, branching principal components and topology preserving mappings are described as well. Presentation of algorithms is supplemented by case studies, from engineering to astronomy, but mostly of biological data: analysis of microarray and metabolite data. The volume ends with a tutorial "PCA and K-means decipher genome". The book is meant to be useful for practitioners in applied data analysis in life sciences, engineering, physics and chemistry; it will also be valuable to PhD students and researchers in computer sciences, applied mathematics and statistics.

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This second, companion volume contains 92 applications developing concepts and theorems presented or mentioned in the first volume.Introductions to and applications in several areas not previously covered are also included such as graded algebras with applications to Clifford algebras and (S)pin groups, Weyl Spinors, Majorana pinors, homotopy, supersmooth mappings and Berezin integration, Noether's theorems, homogeneous spaces with applications to Stiefel and Grassmann manifolds, cohomology with applications to (S)pin structures, Bäcklund transformations, Poisson manifolds, conformal transformations, Kaluza-Klein theories, Calabi-Yau spaces, universal bundles, bundle reduction and symmetry breaking, Euler-Poincaré characteristics, Chern-Simons classes, anomalies, Sobolev embedding, Sobolev inequalities, Wightman distributions and Schwinger functions.

The material included covers an unusually broad area and the choice of problems is guided by recent applications of differential geometry to fundamental problems of physics as well as by the authors' personal interests.Many mathematical tools of interest to physicists are presented in a self-contained manner, or are complementary to material already presented in part I. All the applications are presented in the form of problems with solutions in order to stress the questions the authors wished to answer and the fundamental ideas underlying applications. The answers to the solutions are explicitly worked out, with the rigor necessary for a correct usage of the concepts and theorems used in the book. This approach also makes part I accessible to a much larger audience.

The book has been enriched by contributions from Charles Doering, Harold Grosse, B. Kent Harrison, N.H. Ibragimov and Carlos Moreno, and collaborations with Ioannis Bakas, Steven Carlip, Gary Hamrick, Humberto La Roche and Gary Sammelmann.
... Read more

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We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study. ... Read more

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This reference book, which has found wide use as a text, provides an answer to the needs of graduate physical mathematics students and their teachers. The present edition is a thorough revision of the first, including a new chapter entitled ``Connections on Principle Fibre Bundles'' which includes sections on holonomy, characteristic classes, invariant curvature integrals and problems on the geometry of gauge fields, monopoles, instantons, spin structure and spin connections. Many paragraphs have been rewritten, and examples and exercises added to ease the study of several chapters. The index includes over 130 entries.
... Read more

Customer Reviews (4)

What every young (and not so young!) math phys should know
Writing a review for something that everybody knows its high quality would be a waste of time, but perhaps not anymore - younger people should know the 'standard candles'.

Unless you are in a place where all this material you can attend from lectures, this is the book that if you are (or want to be) a mathematical physicist must try to read 'a little every day', hoping that eventually things will start focusing and you will catch up.

It should be considered in a sense as THE modern analogue of Synge & Schild's Tensor Calculus - it has the same selection of topics but now all on manifolds: Analysis on Manifolds, Riemannian geometry, Integration, Connections, plus distributions and aplications to PDEs and selected topics of infinite-dim geometry.

So you have here a source-book that will not only allow you to formulate, in a modern way, physical laws (differential geometry) but also help you to study them (PDEs).

It is a profitable reading for someone who is somewhat versatile with elementary abstract mathematics, say at the level of Geroch's Mathematical Physics (algebra, topology, measure theory, functional analysis), but once you get going, you never stop!

Start off from chapter 3 and get back (or look up Geroch) if you need an explanation of a word you don't understand or have forgotten.

After you have a basic understanding a Riemannian geometry from this book you'll hopefully be able to reach Mme Choquet's new book on GR and the Einstein Equations, it is a continuation and uses the same notation, or volume 2 of the book under review on various topics in mathematical physics.

Most importantly, keep your cool and don't get intimidated!

an inspiration
This is the book that inspired me as an undergraduate to learn and appreciate, to a large extent, how physics and mathematics cohabit so beautifully. I continue to see, to this day, as a graduate student, how many of the recentdevelopments in theoretical physics have been inspired by new mathematics and conversely. I stillrefer to this book on occasion, since it is laid out in a style that is amenable to mathematicians (such as myself!). It's an excellent read, and I highly recommend it (even as bed time reading!)Best regards, A.

A must have desk reference and educational tool
This was the most reliable (and compact) source of mathematical information that I could find.I used it (almost exclusively) to teach myself what I needed to know to do my PhD thesis in General Relativity withTorsion fields.It is very overwhelming to look at the book the firsttime.But give it time.It is well written and inviting.The only drawback for me was the very small print.

A treasure for physicists
I use this book constantly. At first I thought it was good only as areference, but now I know it is possible to actually LEARN any of itssubjects from scratch. I particularly like its chapters on manifolds, LieGroups and bundles. Connections on a principal bundle is extremely welldone, with a translation to physics (gauge theories) performed in detail inan exercise (so to speak). Some time ago David Ruelle said that thephysicists needed a presentation of recent mathematics in the form ofBourbaki's "Fascicule des resultats", a synthesis of the subjectwith complete definitions, examples and theorems clearly stated, but withthe proofs ommited. This is it, except that in this book mostdemonstrations are not ommited, only those too complicated. The whole bookis extremely readable, if you concentrate, turn out the TV, etc. A preciousbook. ... Read more

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This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in 1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects. ... Read more

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This book provides an easy introduction to the theory of differentiable manifolds. The authors then show how the theory can be used to develop, simply but rigorously, the theory of Lanrangian mechanics directly from Newton's laws. Unnecessary abstraction has been avoided to produce an account suitable for students in mathematics or physics who have taken courses in advanced calculus. ... Read more

Customer Reviews (1)

Probably the Only Book Dedicated to this Subject.

This book is intended to help the young engineer, designer or technician visualize the different possibilities and valve arrangements available for the installation of static pressure measurement and also of differential pressure instruments, used either to measure flow, level, or density.

Since field installations are highly vulnerable to various industry and company practices, historical inertia, individual opinions, and particular or individual standards or regulations, the illustration and written material is useful for discussion and interpretation by every potential reader.

Although a great deal of installation design and layout has been done for all types of applications, that information is isolated in the files of individuals companies. Taking that into account this book is most useful for the non experienced since it appear to be the only book available in this specific topic of proper valve and manifold configuration for different installations and applications of field instruments. ... Read more

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This book is about an investigation of recent developments in the field of sympletic and contact structures on four- and three-dimensional manifolds from a topologist’s point of view. In it, two main issues are addressed: what kind of sympletic and contact structures we can construct via surgery theory and what kind of sympletic and contact structures are not allowed via gauge theory and the newly invented Heegaard-Floer theory.

Customer Reviews (3)

Nice Treatment
Do you know what a Capped grope is? If so, this is the book for you! This is a really readable, though understandably advanced, book. It seems to contain everything a person could possibly want to know about embedding a disk in a 4-manifold. I'm far from an expert on 4-manifolds, but I haven't seen much of this material presented anywhere else. If you want to become a master of 4-manifolds, you need this book like Luke Skywalker needed Yoda.

My copy doesn't have the cool orange cover so I feel a little sad. Hopefully you won't get the brown one like me.

The weirdness of four-manifolds
Lee Carlson's review can hardly be improved upon, but I have one addendum, responding to his lament at the end of the review:

A concrete construction of a fake 4-space can be found in the last section of Chapter 1 of Dan Freed and Karen Uhlenbeck's (1984, revised 1991) book, "Instantons and Four-Manifolds", a great compliment to Freedman and Quinn's book. Unfortunately, it's just as hard to find and about three times as costly as the reviewed book. Unless you really can't live without seeing how this is done, take it on faith (not much consolation to a physicist who wants to know if black holes behave differently in fake spacetime). A cheaper alternative: If you visit a major university math library and take a bunch of quarters for their copier, the relevant Ch.1, "Fake R^4", is fourteen pages long.

Brilliant overview of topology in 4 dimensions
It is too bad this book is out of print, for it introduces the reader to a fascinating branch of topology and has the clearest proof of the 4-dimensional Poincare conjecture. In addition, the authors do not hesitate to employ diagrams as needed to illustrate the main points and to assist the reader in visualizing 4-dimensional objects. The authors give a fine discussion as to the reasons why four dimensions is harder to deal with topologically than dimensions five or greater, this being essentially due to the behavior of 2-dimensional disks: mapping 2-disks into 3-manifolds results (generically) with 1-dimensional self-intersections; in 4-dimensions the intersections are isolated points, and in 5 dimensions or more the 2-disks can be embedded.

Interestingly, the authors choose not to employ the famous "Kirby calculus" in the proofs of the main results, despite the fact that it was used extensively in their earlier works. They break the book into two parts, the first one emphasizing embedding theorems and the second one the structure of manifolds. Those readers interested in the proof of the 4-dimensional Poincare conjecture will find it in chapter 7, as a consequence of the authors proof of the h-cobordism theorem, the latter being nontrivial. It is the absence of a smooth structure on the h-cobordism that makes it so difficult in dimension four.

The existence of exotic structures on 4-manifolds is discussed in detail in chapter 8 and the authors endeavor to show why dimension 4 is unique compared to higher dimensions. The existence of exotic structures on 4-manifolds is definitely interesting, and has recently been shown to have importance in physics. But physicists who need an explicit example of one of these structures will not find one here, and I know of no such examples in the literature. Such an example would be interesting from the standpoint of the behavior of quantum field theories on such 4-manifolds, as one would like to know if this behavior would indeed be different than that on the manifold with the "standard structure". ... Read more