Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.
(source: Nielsen Book Data) 9780821849255 20160604

• Preface 1.1Train Tracks 1.2Multiple Curves and Dehn's Theorem 1.3Recurrence and Transverse Recurrence 1.4Genericity and Transverse Recurrence 1.5Trainpaths and Transverse Recurrence 1.6Laminations 1.7Measured Laminations 1.8Bounded Surfaces and Tracks with Stops Ch. 2Combinatorial Equivalence 2.1Splitting, Shifting, and Carrying 2.2Equivalence of Birecurrent Train Tracks 2.3Splitting versus Shifting 2.4Equivalence versus Carrying 2.5Splitting and Efficiency 2.6The Standard Models 2.7Existence of the Standard Models 2.8Uniqueness of the Standard Models Ch. 3The Structure of ML[subscript 0] 3.1The Topology of ML[subscript 0] and PL[subscript 0] 3.2The Symplectic Structure of ML[subscript 0] 3.3Topological Equivalence 3.4Duality and Tangential Coordinates Epilogue Addendum The Action of Mapping Classes on ML[subscript 0] Bibliography.
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Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry and dynamical systems. This book presents a self-contained treatment of the combinatorial structure of the space of measured geodesic laminations in a fixed surface. Families of measured geodesic laminations are described by specifying a train track in the surface, and the space of measured geodesic laminations is analyzed by studying properties of train tracks in the surface.
(source: Nielsen Book Data) 9780691025315 20160528