Book Description

The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time.

Key Features
* Treats differential geometry, differential topology, and quantum field theory
* Includes elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory
* Tackles problems of quantum field theory using differential topology as a tool ... Read more

Reviews (2)

Fair treatment
This book is written for the theoretical physicist in mind. It is somewhat out-of-date, as there have been many developments in differential topology, such as the Seiberg-Witten theory, since this book was published. However, it might still serve the reader with an introduction to these latter developments. Being just a summary there are no proofs given, and this might annoy the mathematician, but it still could be read profitably by a mathematician if they need a quick introduction to the interplay between physics and differential topology. Indeed, some of the results outlined in the book have been dubbed "physical mathematics" by mathematicians because of the lack of rigour involved in some of the constructions. Quantum field theory is not a subject at all to look for mathematical rigour. Several attempts have been made to put it on a sound mathematical foundation, by these attempts all result in a weakening of its physical predictive power.

The author introduces some basic topological notions in the first chapter, such as homotopy and homology/cohomology groups. He does give a good explanation via the smash product, of how to get a base point in a product space when each factor has a base point. Also, his discussion of H-and coH-spaces is very intuitive and serves the physicist-reader well in developing a functorial mind set. Freedman's and Donaldson's work in 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional topology is discussed only very briefly however. The existence of exotic structures on 4-dimensional Euclidean space has recently been shown to have interesting physical consequences, but the author only devotes a few sentences to exotica. The explicit construction of an exotic structure is of great importance to physical applications, but as of yet only existence results are known.

Physicists are used to dealing with elliptic partial differential equations, and the next chapter discusses these in a more abstract guise: the theory of elliptic operators. These are introduced in the context of vector bundles, as preparation for the Atiyah-Singer index theorem. The locality of pseudo-differential operators sets up the need for Sobolev spaces, and the author does a fairly good job of overviewing the main results.

The concept of a sheaf is introduced in the next chapter, but I think physicists would understand sheaves better if they were introduced via analytic continuation, a procedure that physicists are very well acquainted with. K-theory is also discussed in this chapter and the corresponding stable theory. Physicists have to understand the Bott periodicity theorems when doing functional integration in quantum field theory. Characteristic classes are only briefly treated, and, like all the treatments of this subject, the discussion gives no insight as to why these objects work as well as they do.

The author returns to elliptic operators in the next chapter, where their index theory is discussed. The treatment is too formal. and the reader will have to search the literature formore in-depth discussion.

Algebraic geometry, which has taken on immense importance in string theories and M-theory, is introduced in the next chapter. This chapter might be too quick for the physicist needing an understanding for applicaations in these areas. More concrete examples of varieties and explicit calculations of moduli spaces would have been helpful.

Physicists who have done current algebra will appreciate the next chapter on infinite dimensional groups. The loop group, gauge group, Virasoro group, and the Kac-Moody algebra, of use in conformal field theories and gauge field theories, are given fairly good treatment.

Morse theory, so indispensable in both mechanics and quantum field theory, is discussed in the next chapter. This is probably the best written of the chapters in the book, especially the sections on equivariance and supersymmetry.

Instantons, so important in guage theories and the subsequent quantization via functional integration, are treated in Chapter 8. It is a fairly good discussion, with infinite dimensional critical point theory given emphasis.

Applications to string theory is the subject of the next chapter, but the chapter is far too short to be of much use to someone first entering the field.

The treatment of anomalies in the next chapter is quite good though; the section on Fock space and Gauss's law is one of the best I have seen in the literature. The author explains carefully the origin of the Schwinger term.

Conformal field theories follow in Chapter 9, and the Virasoro algebra again makes its appearance. This is an area that employs more of the "hard" analysis in obtaining results rather than "soft" techniques, so physicists should be fairly comfortable with the discussion.

The last chapter introduces a topic that could fairly be classified as a "quantization of mathematics". The author discusses topological quantum field theories, and it is in this area that I believe the most fascinating constructions in all of mathematical physics take place. These theories have spurred a tremendous amount of research, and the author gives a fairly good overview. The book is a little too overpriced considering the content and the fact that it is a paperback. Such expense is worth it for a self-contained book, but this is not one of these, and must be supplemented by a great deal of outside reading.

A mathematician with a physics background
This book has a huge amount of mathematics packed inside it.It covers most of the math needed for understanding string theory, but because of its scope very few of the theorems are proved. I have found it very helpful as an overview. ... Read more

Book Description

The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of its own and as an approach to control theory. The book deals with the theory of semi-linear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. This assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented. ... Read more

Book Description

This book provides an elementary introduction to the ideas and methods of topology by the detailed study of certain topics. There are elegant but rigorous proofs of many of the basic theorems, and special attention is given to the results needed in the theory of functions. ... Read more

Reviews (1)

A Classic
One of the best introductions on Metric Spaces and topology. Easy to read and learn from. ... Read more

Book Description

This volume presents an array of topics that introduce the reader to key ideas in active areas in geometry and topology. The material is presented in a way that both graduate students and researchers should find accessible and enticing. The topics covered range from Morse theory and complex geometry theory to geometric group theory, and are accompanied by exercises that are designed to deepen the reader's understanding and to guide them in exciting directions for future investigation. ... Read more

Reviews (1)

Excellent book
This short book is a good place to start learning about the theory of Reidemeister torsions. Masterfully written by one of the leading experts on the subject, it follows the standard of exposition set by Milnor in his famous survey on this subject 40 years ago. The book is very easy to read, the necessary background is covered economically, but in details. You can find here many results that received the first treatment in book form and some classical results with illuminating explainations. The last chapter briefly indicates recent developments in connection with Seiberg-Witten invariants of 3-manifolds.

The book is mainly an introduction and concerns mostly with combinatorial manifolds and knot theory. As the author stated, many other aspects of the theory (e.g. the analytical theory) are left out. ... Read more

Reviews (1)

I don't like this book
We used this book for the first undergraduate topology course at my university. Perhaps the book is okay if you know what to expect from it before you buy it. What this book IS NOT is a book on General Topology. What this book IS is a book on the topology of the real number system. It's odd, because the course I took is one on general topology, where the topology of the real number system was mentioned as a mere footnote, yet this book covers very little general topology. At the end of each chapter, there is a starred section, in which about 1 page is devoted to how these concepts apply to general topology. So if that's what you want, perhaps you'll like this book. But that's not the only reason I didn't like it.

This book states EVERYTHING as a proposition. It gets ridiculous, there's so many propositions. In one section of one chapter I counted almost 40 propositions once. Unbelievable. Many results are not proved, which leaves alot to be desired in that department, and alot of the exercises are "give an example of...". These types of problems are great, but all in all there just isn't enough variety in this book for me.

So, if you're looking for a book on General Topology this definitely is not the one you want. If you're looking for a book on Topology of Rn, then I'd say you can consider this book rated 3.5 stars, and you can make your decision based on that.

However, if that's really all you want, you can find the same results in any modern analysis book such as Baby Rudin (i.e. Principles of Mathematical Analysis), and for general topology the book(s) to go to are Munkres' Topology, and Kelley's General Topology. ... Read more

Book Description

This text is intended for a one-semester undergraduate course in topology.The fundamental concepts of general topology are covered rigorously but at a gentle pace and an elementary level.It is accessible to students with only an elementary calculus background. In particular, abstract algebra is not a prerequisite.The first chapter develops the elementary concepts of sets and functions, and in Chapter 2 the general topological space is introduced.Subspaces, continuity, and homeomorphisms are covered in Chapter 3.The remaining chapters cover product spaces, connected spaces, separation properties, and metric spaces. ... Read more

Reviews (3)

Avoid Introduction to Topology by Baker
This books is terrible. There is no answers in the book! You have no way of knowing if you are doing the problems right or not. Also, the examples are too brief and don't explain what steps or theorems that they used. I am buying another book just to figure out what this book is talking about.

Good introduction
Written for undergraduate students of mathematics, this book serves as a fine introduction to topology from an abstract set-theoretic point of view. The approach of the author is also to have the reader do most of the proofs of the theorems in the book, and thus the book can be thought of as the student's second course in proof theory, the first being maybe Euclidean geometry.

Basic concepts in set theory are outlined in chapter 1, and in detail. This paves the way for a discussion of topological spaces in chapter 2. The author could have begun this discussion with a general definition of a topological space, but instead chooses (thankfully) to motivate the definition via the definition of an open set in the real line. The abstract definition is thus better appreciated, which the author then does immediately. He then moves on to the consideration of subspaces and continuity in chapter 3. The discussion here is pretty standard, as there are not any examples that cannot be found in the literature. The most important concept introduced here is that of a homeomorphism, and readers will get a taste of the intuitive "coffee cup = donut" definition of topological equivalance that they have no doubt heard about from popular discussions of topology.

Product spaces are introduced in chapter 4, with a brief peek at topological groups given. Infinite products are introduced but the reading is labeled as supplementary by the author. Chapter 5 is then an introduction to the topological concept of connectedness. The beginner may be troubled as to the way connectedness is defined, since it is defined as the negation of disconnectedness, but the examples given should alleviate any skepticism as to this nonconstructive definition of connectedness. The famous example of a connected space that becomes totally disconnected after the removal of one point is unfortunately not discussed in this book.

Another important concept in topology, that of compactness, is discussed in chapter 6. It is introduced via the concept of coverings, and it is shown that the use of this concept, and not one that is based on a generalization of closed and bounded sets, is the one that gives the best definition for general topological spaces. Then, in chapter 7, the separation properties of topological spaces are discussed. Regular and normal spaces are defined here also, along with the concept of a T5 space. The latter is usually not discussed in elementary books in topology. Metric spaces are finally introduced in the last chapter of the book, giving the reader some of the tools needed for a future study of analysis.

Excellent for introducing abstract mathematics
This book is so attainable to a reader with a solid foundation in mathematics regardless of the complexity and depth of the subject matter.Baker first gives a good review of basic set theory, which I believe is always something that can never be reviewed enough. He then proceeds to beautifully bring together definitions and examples leading, of course, to theorems and corrollaries.This is all backed by exercises that, which do require much work and thought, are very reinforcing and rewarding. I have just begun my graduate career in mathematics and having Baker's book at hand has been a saving grace.Although this book is introductory and therefore lacking in some concepts, it covers the basic material so well that moving forward in the subject of topology becomes accesible and enjoyable. ... Read more

Book Description

This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André Weil, as well as a proof of the Gauss–Bonnet–Chern theorem based on the Mathai–Quillen construction of Thom forms; the second part presents analytic proofs of the Poincaré–Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten. ... Read more

Book Description

For the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology.This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds.Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory.This book is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them.While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology. ... Read more

Book Description

Book Description

Equivariant cohomology in the framework of smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Brning and V. M. Guillemin. The point of departure are two relatively short but very remarkable papers by Henry Cartan, published in 1950 in the Proceedings of the "Colloque de Topologie". These papers are reproduced here, together with a scholarly introduction to the subject from a modern point of view, written by two of the leading experts in the field. This "introduction", however, turns out to be a textbook of its own presenting the first full treatment of equivariant cohomology from the de Rahm theoretic perspective. The well established topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects (like symplectic geometry, Lie theory, dynamical systems, and mathematical physics), leading up to the localization theorems and recent results on the ring structure of the equivariant cohomology. ... Read more

Reviews (1)

From Geometry to Algebra
This book is inspired by Cartan's two seminal papers about the topological implications of Lie groups acting on manifolds.In fact, the two papers are reprinted in the back of the book.

There is not much new mathematicsneeded to extract the category of G^* algebras and to develop theirequivariant cohomology.This abstraction has fascinatingimplications.

There are chapters treating special topics in detail, suchas spectral sequences and fermionic integration.Throughout the book iswell-written and clear.

Any person interested in equivariant cohomologyor symplectic geometry should read this book! ... Read more

Book Description

In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together.The book develops algebraic/differential topology from an application-oriented point of view.The book takes the reader on a path starting from a well-motivated robust stability problem, showing the relevance of the simplicial approximation theorem and how it can be efficiently implemented using computational geometry. The simplicial approximation theorem serves as a primer to more serious topological issues such as the obstruction to extending the Nyquist map, K-theory of robust stabilization, and eventually the differential topology of the Nyquist map, culminating in the explanation of the lack of continuity of the stability margin relative to rounding errors. The book is suitable for graduate students in engineering and/or applied mathematics, academic researchers and governmental laboratories. ... Read more

Book Description

This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals. ... Read more