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Elementary Applied Topology 1st Edition by Robert Ghrist ISBN 1502880857 978-1502880857

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Elementary Applied Topology 1st Edition Robert Ghrist Digital Instant Download

Author(s): Robert Ghrist
ISBN(s): 9781502880857, 1502880857
Edition: 1
File Details: PDF, 17.60 MB
Year: 2014
Language: english
SKU: EB-5145558 Category: Tag:

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ISBN 10:  1502880857

ISBN 13:  978-1502880857

Author: Robert Ghrist

This text gives a brisk and engaging introduction to the mathematics behind the recently established field of Applied Topology. Over a century of development of principles and techniques in algebraic topology has of late crossed over to a variety of application domains. This text gives a completely novel introduction to these methods in the context of the applications. “Elementary Applied Topology” is short (250 pp. plus bibliography and index) and richly illustrated, with 268 figures. It is perfect for both self-study, and as the basis for a course in applied topology. This book is also well-suited for use as a supplementary text in a more traditional algebraic topology course, providing both context and motivation for the tools to be learned.The progression of mathematical techniques is a fresh approach. The book begins with a quick trip through manifolds and cell complexes. The segue to algebraic topology comes in the form of the Euler characteristic and the Euler calculus born from it. Passing from this to homology, exact sequences, and cohomology sets the stage for the innovative content to come. This is comprised of modern Morse theory (including discrete Morse theory, Conley index, and stratified Morse theory), sheaf theory (with an emphasis on cellular sheaves and cosheaves), and, finally, category theory and categorification. Every tool and topic is paired with an application. These range in scope across the biological, economic, engineering, material, physical, and statistical sciences. Of particular note are the applications to topological data analysis, including persistent homology and barcodes. “Elementary Applied Topology” is the first comprehensive text on applied algebraic topology for students of all mathematical sciences.
 

Table of contents: 

1 Manifolds

1.1 Manifolds

1.2 Configuration spaces of linkages.

1.3 Derivatives.

1.4 Vector fields

1.5 Braids and robot motion planning

1.6 Transversality

1.7 Signals of opportunity

1.8 Stratified spaces

2 Complexes

2.1 Simplicial and cell complexes

2.2 Vietoris-Rips complexes and point clouds

2.3 Witness complexes and landmarks

2.4 Flag complexes and networks

2.5 Čech complexes and random samplings.

2.6 Nerves and neurons

2.7 Phylogenetic trees and links

2.8 Strategy complexes and uncertainty.

2.9 Decision tasks and consensus

2.10 Discretized graph configuration spaces

2.11 State complexes and reconfiguration

3 Euler Characteristic

3.1 Counting

3.2 Curvature..

3.3 Nonvanishing vector fields.

3.4 Fixed point index.

3.5 Tame topology

3.6 Euler calculus

3.7 Target enumeration

3.8 From Fubini Thecre

3.9 Euler integral transforms.

3.10 Intrinsic volumes

3.11 Gaussian random fields

4 Homology

4.1 Simplicial and cellular homology.

4.2 Homology examples

4.3 Coefficients.

4.4 Singular homology

4.5 Reduced homology

4.6 Czech homologues.

4.7 Relative homology.

4.8 Local homology

4.9 Homology of a relation

4.10 Functoriality

4.11 Inverse kinematics

4.12 Winding number and degree

4.13 Fixed points and prices

5 Sequences

5.1 Homotopy invariance.

5.2 Exact sequences

5.3 Pairs and Mayer-Vietoris.

5.4 Equivalence of homology theories

5.5 Cellular homology, redux

5.6 Coverage in sensor networks

5.7 Degree and computation

5.8 Borsuk-Ulam theorems

5.9 Euler characteristic.

5.10 Lefschetz index.

5.11 Nash equilibria

5.12 The game of Hex

5.13 Barcodes and persistent homology

5.14 The space of natural images..

5.15 Zigzag persistence

6 Cohomology 

6.1 Duals.

6.2 Cochain complexes.

6.3 Cohomology

6.4 Poincaré duality

6.5 Alexander duality.

6.6 Helly’s Theorem

6.7 Numerical Euler integration

6.8 Forms and Calculus

6.9 De Rham cohomology

6.10 Cup products.

6.11 Currents

6.12 Laplacians and Hodge Theory

6.13 Circular coordinates in datsets.

7 Morse Theory

7.1 Critical points

7.2 Excursion sets and persistence

7.3 Morse homology

7.4 Definable Euler integration

7.5 Stratified Morse theory

7.6 Conley index

7.7 Lefschetz index, redux

7.8 Discrete Morse thecry

7.9 LS category

7.10 Unimodal decomposition in statistics

8 Homotopy

8.1 Group fundamentals

8.2 Covering spaces

8.3 Knct theory

8.4 Higher homotopy groups.

8.5 Biaxial nematic liquid crystals

8.6 Homology and homotopy

8.7 Topological social choice

8.8 Bundles..

8.9 Topological complexity of path planning

8.10 Fibrations..

8.11 Homotopy type thecry

9 Sheaves

9.1 Cellular sheaves

9.2 Examples of cellular sheaves.

9.3 Cellular sheaf cohomology

9.4 Flow sheaves and obstructions

9.5 Information flows and network coding

9.6 From cellular to topological

9.7 Operations on sheaves..

9.8 Sampling and reconstruction

9.9 Euler integration, redux

9.10 Cosheaves

9.11 Bézier curves and splines

9.12 Barcodes, redux

10 Categorification

10.1 Categories

10.2 Morphisms

10.3 Functors

10.4 Clustering functors

10.5 Natural transformations

10.6 Interleaving and stability in persistence

10.7 Limits.

10.8 Colimits

10.9 Sheaves, redux

10.10 The genius of categorification

10.11 “Bring out number”

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