000			01555nam a2200193 4500
020			_a9781470438463
082			_a515 _bROE/W
100			_aRoe, John
245			_aWinding around : _bthe winding number in topology, geometry, and analysis
260			_aRhode Island _bAmerican mathematical society _c2015
300			_a269 p.
520			_aThe 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).
650			_aMathematical analysis--Foundations
650			_aSymmetric functions
650			_aAssociative law (Mathematics)
650			_aCommutative law (Mathematics)
650			_aAlgebraic topology
942			_cBK
999			_c67058 _d67058