William F. Basener

Topology and Its Applications (eBook, PDF)

• Format: PDF

Produktbeschreibung

Discover a unique and modern treatment of topology employing a cross-disciplinary 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 real-world 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 point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include: * Open sets * Compactness * Homotopy * Surface classification * Index theory on surfaces * Manifolds and complexes * Topological groups * The fundamental group and homology 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.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 384
• Erscheinungstermin: 18. Oktober 2011
• Englisch
• ISBN-13: 9780470067932
• Artikelnr.: 37290029

Autorenporträt

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.

Inhaltsangabe

Preface. 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
path-connectedness. 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 Borsuk-Ulam 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
n-Symmetry 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
Seifert-van Kampen Theorem. 5.4.1 Flowers and Surfaces. 5.4.2 The
Seifert-van 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 A-complexes. 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.

Rezensionen

"..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)

"...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)