Book Description

This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover. ... Read more

Reviews (2)

N. Bourbaki is the greatest mathematician ever!! Well,...
After reading the story of N. Bourbaki, I HAD to order every book by Bourbaki that I could find.All I'll say is I love the book, but perhaps even more I love the concept of Bourbaki (if you've never heard of Bourbaki, google it for an interesting read).

the book that the experts study
i recently ordered "A GENERAL TOPOLOGY WORKBOOK" by Iain Adamson, as i very much wish to understand the mathematics of topology.as i would do with any other book, i started by reading the introduction and suggested readings.Mr. Adamson highly recommends the BOURBAKI series on topology as a reference material.he ascribes to this series as his text of choice and further states that this is the text that he has studied the most closely.it is for this reason that i am ordering this series.

i extend my thanks to mr. adamson for the recommendation.with the plethora of choices in study materials and a limited budget, i needed to narrow my scope and decide which text would best serve me.

thanks again:-) ... Read more

Book Description

This book covers topology and geometry beginning with an accessible account of the extraordinary and rather mysterious impact of mathematical physics, especially gauge theory, on the study of the geometry and topology of manifolds.Much of the mathematics developed in the book to study the classical field theories of physics (de Rham cohomology, Chern classes, Semi-Riemannian manifolds, Cech cohomology, spinors etc. ) is standard, but the treatment always keeps one eye on the physics and unhesitatingly sacrifices generality to clarity.The author brings the reader up to the level needed to conclude with a brief discussion of the Seiberg-Witten invariants. Although this volume can be read independently Naber carries on the program initiated in his earlier volume, Topology, Geometry and Gauge Fields: Foundations, Springer, 1997, and writes in much the same spirit with precisely the same philosophical motivation. A large number of exercises are included to encourage active participation on the part of the reader.This work will be of great interest to researchers and graduate students in the field of mathematical physics. REVIEWS OF TOPOLOGY, GEOMETRY, AND GAUGE FIELDS: FOUNDATIONS"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics. . . Naber combines a knowledge of his subject with an excellent informal writing style. "NZMS NEWSLETTER". . . this book should be very interesting for mathematicians and physicists (as well as other scientists) who ae concerned with gauge theories. "ZENTRALBLATT FUER MATHEMATIK ... Read more

Reviews (2)

MATH AND TOPOLOGY
Topology is very important scince in the fields of mathematics. And it using in many of another sinceis.

required reading for a topologist interested in physics
As a mathematician turned physics grad student, it is often difficult to read "Math for Physicists" books simply because of the focus on making "numbers churn out;" which, at least for me personally, more difficult to get a handle on the subject and then, in turn, use it fruitfully.

This book on the other hand, is exemplary of why I got into physics in the first place. The first chapter (Physical motivations) and the last chapter (Gauge Fields and Instantons) can be read by any one with undergraduate topology under their belt and come away with a more powerful understanding of gauge theory than, in my opinion, can be found in other introductory gauge theory texts I've been directed to.

Of course I'll read all those said texts as well, but I'm thankful that I found this one. ... Read more

Book Description

This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981). This second edition is a brand new book and completely supersedes the original version of nearly 30 years ago. But a lot of the material has been rearranged, rewritten, or replaced by a more up-to-date exposition, and a good deal of new material has been incorporated in this book, all reflecting the progress made in the field during the last three decades. ... Read more

Book Description

This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level.The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory.These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery.Wherever possible, the geometric motivation behind the various concepts is emphasized.The text consists of material from the first five chapters of the author's earlier book, ALGEBRAIC TOPOLOGY: AN INTRODUCTION (GTM 56), together with almost all of the now out-of-print SINGULAR HOMOLOGY THEORY (GTM 70).The material from the earlier books has been carefully revised, corrected, and brought up to date. ... Read more

Reviews (1)

Excellent text on algebraic topology
The text contains material from the author's earlier two books; Algebraic Topology: An Introduction (GTM 56), and Singular Homology Theory (GTM 70). The book starts with an introductory chapter on 2-manifolds and then continues with the fundamental group; which is conceptually easier than homology, with which some books on algebraic topology start. The only prerequisite for this book is a basic knowledge of general topology; and the book is easily accessible to anyone studying on his own. In short, I recommend the book to anyone interested in algebraic topology. ... Read more

Book Description

Boiled-down essentials of the top-selling Schaum's Outline series for the student with limited time

What could be better than the bestselling Schaum's Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum's Easy Outline series. Every book in this series is a pared-down, simplified, and tightly focused version of its predecessor. With an emphasis on clarity and brevity, each new title features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form.

Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give students quick pointers to the essentials.

• Designed to appeal to underprepared students and readers turned off by dense text
• Cartoons, sidebars, icons, and other graphic pointers get the material across fast
• Concise text focuses on the essence of the subject
• Delivers expert help from teachers who are authorities in their fields
• Perfect for last-minute test preparation
• So small and light that they fit in a backpack!

Book Description

In this superb topology text, the readers not only learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, but also their role in understanding molecular structures.Most results described in the text are motivated by the questions of chemists or molecular biologists, though they often go beyond answering the original question asked.No specific mathematical or chemical prerequisites are required. The text is enhanced by nearly 200 illustrations and 100 exercises. With this fascinating book, undergraduate mathematics students escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists find simple and clear but rigorous definitions of mathematical concepts they handle intuitively in their work. ... Read more

Reviews (1)

How can topology be applied to chemistry?
This question is answered in a remarkable new book by Erica Flapan, knottheorist and professor of mathematics at Pomona College.Stereochemistry,the study of the three-dimensional structure of molecules, is a recurringtheme in chemistry and related fields.Topology, the study of geometricalproperties which are invariant under continuous transformations, is asimilarly popular area for mathematicians.While it is not immediatelyobvious that the two fields have anything in common, both fields owe a debtto the other.Although she initiates her book with a historicalperspective and detailed expository on basic aspects of low dimensionaltopology (e.g., stereoisomers, chirality, nonrigid symmetries, knot andlink types, three-dimensional manifolds, and link polynomials) she doesproceed into more advanced subject matter in later chapters includingMöbius ladders, symmetries of embedded graphs, and hierarchies ofautomorphisms.Of particular interest to the molecular biologist, thefinal chapter is devoted entirely to the topology of DNA and includestopological considerations of supercoiling, toroidal winding, enzymeaction, and tangle theory. The arguments are clearly presented within theframework of interesting and relevant molecular structures, yet there isenough mathematical rigor to satisfy dyed-in-the-wool mathematicians aswell.Although there will likely be something of interest to the averageworking chemist, the supramolecular scientist, organic chemist, molecularbiologist, and biophysicist, in particular, stand to gain the most by thecontents of this book. ... Read more

Book Description

"The appearance of Grünbaum's book Convex Polytopes in 1967 was a moment of grace to geometers and combinatorialists. The special spirit of the book is very much alive even in those chapters where the book's immense influence made them quickly obsolete. Some other chapters promise beautiful unexplored land for future research. The appearance of the new edition is going to be another moment of grace. Kaibel, Klee and Ziegler were able to update the convex polytope saga in a clear, accurate, lively, and inspired way." --Gil Kalai, The Hebrew University of Jerusalem "The original book of Grünbaum has provided the central reference for work in this active area of mathematics for the past 35 years...I first consulted this book as a graduate student in 1967; yet, even today, I am surprised again and again by what I find there. It is an amazingly complete reference for work on this subject up to that time and continues to be a major influence on research to this day." --Louis J. Billera, Cornell University "The original edition of Convex Polytopes inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." --Peter McMullen, University College London

The combinatorial study of convex polytopes is today an extremely active and healthy area of mathematical research, and the number and depth of its relationships to other parts of mathematics have grown astonishingly since Convex Polytopes was first published in 1966.The new edition contains the full text of the original and the addition of notes at the end of each chapter. The notes are intended to bridge the thirty five years of intensive research on polytopes that were to a large extent initiated, guided, motivated and fuelled by the first edition of Convex Polytopes. The new material provides a direct guide to more than 400 papers and books that have appeared since 1967. Branko Grünbaum is Professor of Mathematics at the University of Washington. ... Read more

Reviews (3)

More Rigorous Than Some Introductory Texts
I purchased Introduction to Metric and Topological Spaces two years ago. I was unprepared for its rigor. I am not a mathematics major, but I enjoy reading mathematics. My background includes calculus, linear algebra, differential equations, and other applied mathematics, but I have not had a course in real analysis. W. A. Sutherland intended this text as the next step after analysis.

After a brief foray, I retreated, placed Sutherland back on my bookshelf, and attacked some marginally easier introductory texts: Metric Spaces by Victor Bryant, Introduction to Topology by Bert Mendelson, and most recently, several chapters in Introduction to Analysis by Maxwell Rosenlicht. I periodically return to W. A. Sutherland's text to measure my understanding. I am now working on chapter five, Compact Spaces.

I doubt that Introduction to Metric and Topological Spaces would be foreboding to students that are familiar with real analysis. Sutherland understands that the abstractness and generalization can be difficult and shows concern with motivating the student. He repeatedly attempts to illustrate the value of generalization, especially in the study of continuity.

Sutherland often uses a lengthy series of examples of increasing difficulty to illustrate abstract concepts. In his discussion of metric spaces, we begin with Euclidian n-space metrics, and move on to discrete metric spaces, function spaces, and even Hilbert sequence spaces. He introduces open sets and topological spaces in a similar fashion.

The author occasionally suggests that the student might wish to make a geometrical diagram to help clarify some subtle point, but Sutherland includes few geometrical drawings in his text. His focus is clearly on proofs using the axioms of metric spaces and topological spaces.

Sutherland highlights sections that either require more knowledge of abstract algebra, or for other reasons are thought to be more severe.

Despite Sutherland's use of Introduction in the title, I suggest that any reader considering independent study might defer tackling Introduction to Metric and Topological Spaces until after completing a more basic text. Possibly a better title might be A Second Introduction to Metric and Topological Spaces.

An excellent, concise book on topology and metric spaces.
I enjoyed reading this book because of its clarity, conciseness, and nice way of relating topological and metric spaces. This book is ideal for the student who is learning about these subjects for the first time, whether or not they intend to do more advanced work on the subject. The reader who wants to go on and learn about more advanced topics, should consult Munkres's book.

A great self-contained text
A lot of books on topology assume some basic knowledge of real analysis, which can throw a lot of readers off. This book starts from the very beginning, and thus is truly a great introduction. Each section has some good exercises, with even a few pointers at the back of the book for the more challenging ones. It starts with topological aspects, and then refers to them in the case of metric spaces (amongst many others), which is a much better approach than most other books, as the reader doesn't take the details of the specific to the general. A great little book, which is a must for most advanced maths Analysis courses. ... Read more

Book Description

The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology--two important ingredients of the mirror symmetry conjecture.

This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds. ... Read more

Reviews (2)

A good reference/second book on knot theory
I don't feel that this book would be the best systematic introduction to the subject (say, for a course on knot theory).However, once someone has been introduced to knot theory(say, via a topology of manifolds class, more elementary book such as Adams, Livingston, or even a more advanced book such as Zieshang-Burde, Lickorish or Rolfsen), this book is an excellent reference.

The strength of this book is the "hands on" explinations given about many of the standard topics on knot theory (Alexander polynomial, Skein invariants, covering spaces, etc.) and I feel that the author does a great job on relating many of the combinatorial invariants to the topology of the knot complement.Many informative illustrations and examples are provided. This is one of the first references I look to when I need a refresher on a topic, or if I encounter something in classical knot theory that I am unfamiliar with.

Also, this book is just plain fun to read!

Of course, this book is from the mid 80's and therefore does not cover some of the more modern material.

Frankly, I've found that anything written by professor Kauffman to be well written and worth reading.

Good intro to knot theory, with a lot of technical detail
Starting out with the basics, Kauffman moves on quickly to more difficult concepts using advanced math. The book has a great section on knot tricks, and a nice table of knots. ... Read more

Book Description

Most, yet not all, chemical substances consist of molecules. The fact that molecules have a "structure" is known since the middle of the 19th century. Since then, one of the principal goals of chemistry is to establish the relationships between the chemical and physical properties of substance and the structure of the corresponding molecules. Countless results along these lines have been obtained along these lines and presented in different publications in this field.One group uses so-called topological indices. About 20 years ago, there were dozens of topological indices, but only a few with noteworthy chemical applications. Over time, their numbers have increased enormously. At this moment here is no theory that could serve as a reliable guide for solving this problem. This book is aimed at giving a reasonable comprehensive survey of the present, fin de siècle, state of art theory and practice of topological indices. ... Read more

Book Description

This is the softcover reprint of the English translation of 1996 (available from Springer since 1989) of the later chapters of Bourbaki's 'Topologie générale'. It completes the treatment of general topology begun in Part I (Ch. 1-4, also available in English in softcover). The real numbers having been introduced in Ch. 4, the first chapters of this volume study subgroups and quotients of R (with applications to the 'measurement of magnitudes' and to the log and exp functions), then real vector spaces and projective spaces, then the additive groups Rn (subgroups, quotients, homomorphisms, infinite sums and products). Analogous properties are then studied for complex numbers, in Ch.8. Chapter 9 illustrates the use of real numbers in general topology, studying different important kinds of topological spaces: uniformizable, metric, normal Baire, Polish, Borel spaces.The final chapter deals with the various topologies of function spaces,ending with a section on approximation of functions. ... Read more

Book Description

Contains a wealth of information previously scattered in research journals, conference proceedings and technical reports. Identifies more than 200 unsolved problems. Every problem is stated in a self-contained, extremely accessible format, followed by comments on its history, related results and literature. The book will stimulate research and help avoid efforts on solving already settled problems. Each chapter concludes with a comprehensive list of references which will lead readers to original sources, important contributions and other surveys. ... Read more

Reviews (1)

A concise catalog of theorems and conjectures.
This is a highly technical book that gathers together in one medium-sized volume (less than 300 pages) hundreds of new and classical theorems and conjectures on every conceivable type of graph coloring problem. It is primarily a mathematical treatment in Theorem-Proof style, although the proofs are left to the references (otherwise the book would have been enormous). The book consists of many short sections, each one often less than a page, formally stating a theorem or a conjecture, then briefly summarizing what is known about it, and where the results have been published. As the authors state in the preface, "We did not intend to write a textbook to be read from beginning to end, but rather a catalog suitable for browsing". It seems they have achieved what was intended; it's quite interesting to simply browse through this book.

To give you an idea of the level of the discussion in the text, here is an excerpt from page 1: After a terse definition of vertex coloring and "chromatic number", the authors state that "The existence of the chromatic number follows from the Well-Ordering Theorem of set theory... However, even if it is not assumed that every set has a well-ordering, but maintaining the property that every set has a cardinality, then the statement 'Any finite or infinite graph has a chromatic number' is equivalent to the Axiom of Choice...". If you are unfamiliar with concepts such as well-ordering or the axiom of choice, such a discussion will be of little value to you. However, if you are familiar with these ideas, you will appreciate how quickly the authors jump into meaty discussions. As another example, the chapter on planar graphs begins with a number of excellent questions: "Does there exist a short proof of the four-color theorem...?", "Is there a short argument to demonstrate that the four-color problem is a finite problem?", and "Is there a short argument that proves the existence of a polynomial algorithm to decide if a given planar graph if 4-colorable?", to mention three. The chapter then proceeds to discuss what is currently know about these problems, and many others.

I suspect that the book would be of great interest not only to mathematicians but also computer scientists, as there are numerous discussions/problems on computational complexity. For example, "Does there exist a function g and a polynomial algorithm that for any given input graph G will find a number s, such that the chromatic number of G satisfies s <= X(G) <= g(s)?" (Here X(G) is the chromatic number of graph G.) The authors state that the question was answered affirmatively by Alon in 1993 if X(G) is replaced by "list-chromatic number". This is typical of the problems cataloged in this book: a terse but formally correct statement of a problem followed by what is currently know, with full citations.

The bibliography at the end of each section is extensive, if not daunting, so there should be little problem looking up all relevant literature concerning a given problem. The authors have also set up an on-line archive for up-to-the-minute research results on these problems. This is an excellent reference for those who are interested in serious research in graph coloring. I would not recommend it to undergraduates in computer science or mathematics, nor to those seeking accessible discussions of classic graph algorithms; this is not an introductory text. ... Read more

Book Description

The goal of this book is to explore some of the connections between control theory and geometric mechanics; that is, control theory is linked with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations and in particular with the theory of mechanical systems subject to motion constraints. The synthesis of topics is appropriate as there is a particularly rich connection between mechanics and nonlinear control theory. The aim is to provide a unified treatment of nonlinear control theory and constrained mechanical systems that incorporates material that has not yet made its way into texts and monographs. This book is intended for graduate students who wish to learn this subject and researchers in the area who want to enhance their techniques. ... Read more

Book Description