Winding Around: The Winding Number in Topology, Geometry, and Analysis (Student Mathematical Library)

Main subject categories: • Winding number • Winding number degree • Topology • Geometry • Homotopy2010 Mathematics Subject Classification. • Primary • 55M25 Degree, winding number • Secondary 55M05 Duality in algebraic topology • 47A53 (Semi-) Fredholm operators; index theories • 58A10 Differential forms in global analysis • 55N15 Topological K-theoryThe 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 The winding number is one of the most basic invariants in topology. It measures the number of times a moving point PP goes around a fixed point QQ, provided that PP travels on a path that never goes through QQ and that the final position of PP 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 Prelude: Love, Hate, And Exponentials -- Paths And Homotopies -- The Winding Number -- Topology Of The Plane -- Integrals And The Winding Number -- Vector Fields And The Rotation Number -- The Winding Number In Functional Analysis -- Coverings And The Fundamental Group -- 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. John Roe. Includes Bibliographical References And Index. Content: * Prelude: Love, hate, and exponentials* Paths and homotopies* The winding number* Topology of the plane* Integrals and the winding number* Vector fields and the rotation number* The winding number in functional analysis* Coverings and the fundamental group* Coda: The Bott periodicity theorem* Linear algebra* Metric spaces* Extension and approximation theorems* Measure zero* Calculus on normed spaces* Hilbert space* Groups and graphs* Bibliography* Index Please Read Brand New, International Softcover Edition, Printed in black and white pages, minor self wear on the cover or pages, Sale restriction may be printed on the book, but Book name, contents, and author are exactly same as Hardcover Edition. Fast delivery through DHL/FedEx express.