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Winding Around: The Winding Number in Topology, Geometry, and Analysis

John Roe Pennsylvania State University, State College, PA
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Softcover ISBN: 978-1-4704-2198-4
Product Code: STML/76
List Price: $52.00 Individual Price:$41.60
This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available.
Expected availability date: February 28, 2023
Electronic ISBN: 978-1-4704-2625-5
Product Code: STML/76.E
List Price: $49.00 Individual Price:$39.20
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List Price: $78.00 This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available. Expected availability date: February 28, 2023 Click above image for expanded view Winding Around: The Winding Number in Topology, Geometry, and Analysis John Roe Pennsylvania State University, State College, PA Available Formats:  Softcover ISBN: 978-1-4704-2198-4 Product Code: STML/76  List Price:$52.00 Individual Price: $41.60 This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available. Expected availability date: February 28, 2023  Electronic ISBN: 978-1-4704-2625-5 Product Code: STML/76.E  List Price:$49.00 Individual Price: $39.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$78.00
This item is temporarily out of stock. Order now and your item will ship as soon as stock becomes available.
Expected availability date: February 28, 2023
• Book Details

Student Mathematical Library
Volume: 762015; 269 pp
MSC: 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 far-reaching 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 final-year undergraduate courses.

This book is published in cooperation with Mathematics Advanced Study Semesters.

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

• 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 university-level mathematics for a living often find themselves reading lots of books on the subject. But even for the book-lovers 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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Volume: 762015; 269 pp
MSC: 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 far-reaching 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 final-year undergraduate courses.

This book is published in cooperation with Mathematics Advanced Study Semesters.

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 university-level mathematics for a living often find themselves reading lots of books on the subject. But even for the book-lovers 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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