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Other titles in the Pure and Applied Mathematics: A WileyInterscience Series of series:
Pure and Applied Mathematics: A WileyInterscience Series of #70: Topology and Its Applicationsby William F. Basener
Synopses & ReviewsPublisher Comments:Discover a unique and modern treatment of topology employing a crossdisciplinary approach
Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research. Filling a gap between the teaching of topology and its modern uses in realworld phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs. This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation. The author presents a diversity of topological areas, including pointset topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:
Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site. Book News Annotation:With its ability to categorize and count objects using approximate qualitative information as opposed to exact values, thereby allowing researchers to better understand an array of diverse topics, topology appeals to a wide range of disciplines, including engineering as well as physical and biological sciences. Basener (mathematics and statistics, Rochester Institute of Technology) keeps this diversity in mind in this text for a first course in topology or geometric topology. Working from rigorous theorems and proofs, and offering a broad array of examples and applications he covers point set topology, combinatorial topology, differential topology, geometric topology and algebraic topology in chapters on continuity, compactness and connectedness, manifolds and complexes, homotopy and the winding number, fundamental group, and homology. The applications are fascinating, and include a simple example of chaos, the topology of the universe, vector fields on surfaces, order and emergent patterns in condensed matter physics and computing Betti numbers.
Annotation ©2006 Book News, Inc., Portland, OR (booknews.com) Book News Annotation:With its ability to categorize and count objects using approximate qualitative information as opposed to exact values, thereby allowing researchers to better understand an array of diverse topics, topology appeals to a wide range of disciplines, including engineering as well as physical and biological sciences. Basener (mathematics and statistics, Rochester Institute of Technology) keeps this diversity in mind in this text for a first course in topology or geometric topology. Working from rigorous theorems and proofs, and offering a broad array of examples and applications he covers point set topology, combinatorial topology, differential topology, geometric topology and algebraic topology in chapters on continuity, compactness and connectedness, manifolds and complexes, homotopy and the winding number, fundamental group, and homology. The applications are fascinating, and include a simple example of chaos, the topology of the universe, vector fields on surfaces, order and emergent patterns in condensed matter physics and computing Betti numbers. Annotation Â©2006 Book News, Inc., Portland, OR (booknews.com)
Synopsis:Modern applications of topology have played an important role solving a diverse spectrum of applied problems. In this text serious attention is given to recent applications of topology in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, population modeling and other areas of science and engineering. Most applications are presented in optional sections, allowing an instructor to customize the presentation.
Synopsis:Discover a unique and modern treatment of topology employing a crossdisciplinary approach
Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research. Filling a gap between the teaching of topology and its modern uses in realworld phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs. This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation. The author presents a diversity of topological areas, including pointset topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:
Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site. Synopsis:WILLIAM F. BASENER, PhD, is an Assistant Professor in the Department of Mathematics and Statistics at Rochester Institute of Technology in Rochester, New York. He received his PhD in mathematics in 2001 from Boston University. His research interests include dynamical systems, differential equations, applied topology, economics, and the topology of manifolds. Dr. Basener is the recipient of numerous teaching awards.
About the Author"…helpful to a beginning student, especially one who is interested in the connections between topology and the world of applications." (Mathematical Reviews, 2007k)
"…a welcome addition to what is now a long list of good undergraduate topology books." (CHOICE, August 2007) "..a celebration of topology and its many applications. I enjoyed reading it and believe that it would be an interesting textbook from which to learn." (MAA Reviews, January 12, 2007) Table of ContentsPreface.
Introduction. I. 1 Preliminaries. 1.2 Cardinality. 1. Continuity. 1. 1 Continuity and Open Sets in R^{n}. 1.2 Continuity and Open Sets in Topological Spaces. 1.3 Metric, Product, and Quotient Topologies. 1.4 Subsets of Topological Spaces. 1.5 Continuous Functions and Topological Equivalence. 1.6 Surfaces. 1.7 Application: Chaos in Dynamical Systems. 1.7.1 History of Chaos. 1.7.2 A Simple Example. 1.7.3 Notions of Chaos. 2. Compactness and Connectedness. 2.1 Closed Bounded Subsets of R. 2.2 Compact Spaces. 2.3 Identification Spaces and Compactness. 2.4 Connectedness and pathconnectedness. 2.5 Cantor Sets. 2.6 Application: Compact Sets in Population Dynamics and Fractals. 3. Manifolds and Complexes. 3.1 Manifolds. 3.2 Triangulations. 3.3 Classification of Surfaces. 3.3.1 Gluing Disks. 3.3.2 Planar Models. 3.3.3 Classification of Surfaces. 3.4 Euler Characteristic. 3.5 Topological Groups. 3.6 Group Actions and Orbit Spaces. 3.6.1 Flows on Tori. 3.7 Applications. 3.7.1 Robotic Coordination and Configuration Spaces. 3.7.2 Geometry of Manifolds. 3.7.3 The Topology of the Universe. 4. Homotopy and the Winding Number. 4.1 Homotopy and Paths. 4.2 The Winding Number. 4.3 Degrees of Maps. 4.4 The Brouwer Fixed Point Theorem. 4.5 The BorsukUlam Theorem. 4.6 Vector Fields and the Poincare' Index Theorem. 4.7 Applications I. 4.7.1 The Fundamental Theorem of Algebra. 4.7.2 Sandwiches. 4.7.3 Game Theory and Nash Equilibria. 4.8 Applications 1I: Calculus. 4.8.1 Vector Fields, Path Integrals, and the Winding Number. 4.8.2 Vector Fields on Surfaces. 4.8.3 1ndex Theory for nSymmetry Fields. 4.9 Index Theory in Computer Graphics. 5. Fundamental Group. 5. I Definition and Basic Properties. 5.2 Homotopy Equivalence and Retracts. 5.3 The Fundamental Group of Spheres and Tori. 5.4 The Seifertvan Kampen Theorem. 5.4.1 Flowers and Surfaces. 5.4.2 The Seifertvan Kampen Theorem. 5.5 Covering spaces. 5.6 Group Actions and Deck Transformations. 5.7 Applications. 5.7.1 Order and Emergent Patterns in Condensed Matter Physics. 6. Homology. 6.1 Acomplexes. 6.2 Chains and Boundaries. 6.3 Examples and Computations. 6.4 Singular Homology. 6.5 Homotopy Invariance. 6.6 Brouwer Fixed Point Theorem for D^{n}. 6.7 Homology and the Fundamental Group. 6.8 Betti Numbers and the Euler Characteristic. 6.9 Computational Homology. 6.9.1 Computing Betti Numbers. 6.9.2 Building a Filtration. 6.9.3 Persistent Homology. Appendix A: Knot Theory. Appendix B: Groups. Appendix C: Perspectives in Topology. C.1 Point Set Topology. C.2 Geometric Topology. C.3 Algebraic Topology. C.4 Combinatorial Topology. C.5 Differential Topology. References. Bibliography. Index. What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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