Winding around : the winding number in topology, geometry, and analysis

By: Roe, JohnMaterial type: TextTextPublication details: Rhode Island American mathematical society 2015Description: 269 pISBN: 9781470438463Subject(s): Mathematical analysis--Foundations | Symmetric functions | Associative law (Mathematics) | Commutative law (Mathematics) | Algebraic topologyDDC classification: 515 Summary: 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 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).
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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 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 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).

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