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Product Code:  STML/76 
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Electronic ISBN:  9781470426255 
Product Code:  STML/76.E 
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Book DetailsStudent Mathematical LibraryVolume: 76; 2015; 269 ppMSC: Primary 55; Secondary 57; 47; 58;
The winding number is one of the most basic invariants in topology. It measures the number of times a moving point \(P\) goes around a fixed point \(Q\), provided that \(P\) travels on a path that never goes through \(Q\) and that the final position of \(P\) is the same as its starting position. This simple idea has farreaching applications. The reader of this book will learn how the winding number can
 help us show that every polynomial equation has a root (the fundamental theorem of algebra),
 guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem),
 explain why every simple closed curve has an inside and an outside (the Jordan curve theorem),
 relate calculus to curvature and the singularities of vector fields (the Hopf index theorem),
 allow one to subtract infinity from infinity and get a finite answer (Toeplitz operators),
 generalize to give a fundamental and beautiful insight into the topology of matrix groups (the Bott periodicity theorem).
All these subjects and more are developed starting only from mathematics that is common in finalyear undergraduate courses.This book is published in cooperation with Mathematics Advanced Study Semesters.ReadershipUndergraduates and beginning graduate students interested in (and trying to learn) ideas concentrated around the notion of the winding number and its appearance in such areas of mathematics as analysis, differential geometry, and topology.

Table of Contents

Chapters

Chapter 1. Prelude: Love, hate, and exponentials

Chapter 2. Paths and homotopies

Chapter 3. The winding number

Chapter 4. Topology of the plane

Chapter 5. Integrals and the winding number

Chapter 6. Vector fields and the rotation number

Chapter 7. The winding number in functional analysis

Chapter 8. Coverings and the fundamental group

Chapter 9. Coda: The Bott periodicity theorem

Appendix A. Linear algebra

Appendix B. Metric spaces

Appendix C. Extension and approximation theorems

Appendix D. Measure zero

Appendix E. Calculus on normed spaces

Appendix F. Hilbert space

Appendix G. Groups and graphs


Additional Material

Reviews

This book covers a lot of ground. But it does so in a clear and careful manner that would make a terrific read for the prepared undergraduate. It is a study in how an intuitive idea can transport one into some deep waters of mathematics, and that is an important story to tell.
John McCleary, Mathematical Reviews 
People who teach universitylevel mathematics for a living often find themselves reading lots of books on the subject. But even for the booklovers among us, after you've just read about ten linear algebra texts, all of which look like they were stamped from the same cookie cutter, the process can occasionally wear thin. It's very pleasant, then, to stumble across a book that is genuinely unique, that addresses a topic in a way not found elsewhere, and that teaches you something that you didn't know before. It's even nicer when the book in question does a really good job of it, as is the case with the book under review. ...Roe's writing style is succinct, but clear and quite elegant; I could practically hear a British accent as I read the book. This clarity of writing and the numerous appendices help make the book accessible.
MAA Online


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The winding number is one of the most basic invariants in topology. It measures the number of times a moving point \(P\) goes around a fixed point \(Q\), provided that \(P\) travels on a path that never goes through \(Q\) and that the final position of \(P\) is the same as its starting position. This simple idea has farreaching applications. The reader of this book will learn how the winding number can
 help us show that every polynomial equation has a root (the fundamental theorem of algebra),
 guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem),
 explain why every simple closed curve has an inside and an outside (the Jordan curve theorem),
 relate calculus to curvature and the singularities of vector fields (the Hopf index theorem),
 allow one to subtract infinity from infinity and get a finite answer (Toeplitz operators),
 generalize to give a fundamental and beautiful insight into the topology of matrix groups (the Bott periodicity theorem).
All these subjects and more are developed starting only from mathematics that is common in finalyear undergraduate courses.
Undergraduates and beginning graduate students interested in (and trying to learn) ideas concentrated around the notion of the winding number and its appearance in such areas of mathematics as analysis, differential geometry, and topology.

Chapters

Chapter 1. Prelude: Love, hate, and exponentials

Chapter 2. Paths and homotopies

Chapter 3. The winding number

Chapter 4. Topology of the plane

Chapter 5. Integrals and the winding number

Chapter 6. Vector fields and the rotation number

Chapter 7. The winding number in functional analysis

Chapter 8. Coverings and the fundamental group

Chapter 9. Coda: The Bott periodicity theorem

Appendix A. Linear algebra

Appendix B. Metric spaces

Appendix C. Extension and approximation theorems

Appendix D. Measure zero

Appendix E. Calculus on normed spaces

Appendix F. Hilbert space

Appendix G. Groups and graphs

This book covers a lot of ground. But it does so in a clear and careful manner that would make a terrific read for the prepared undergraduate. It is a study in how an intuitive idea can transport one into some deep waters of mathematics, and that is an important story to tell.
John McCleary, Mathematical Reviews 
People who teach universitylevel mathematics for a living often find themselves reading lots of books on the subject. But even for the booklovers among us, after you've just read about ten linear algebra texts, all of which look like they were stamped from the same cookie cutter, the process can occasionally wear thin. It's very pleasant, then, to stumble across a book that is genuinely unique, that addresses a topic in a way not found elsewhere, and that teaches you something that you didn't know before. It's even nicer when the book in question does a really good job of it, as is the case with the book under review. ...Roe's writing style is succinct, but clear and quite elegant; I could practically hear a British accent as I read the book. This clarity of writing and the numerous appendices help make the book accessible.
MAA Online