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Invitation to Computational Homotopy [Kietas viršelis]

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(Professor of Mathematics, National University of Ireland, Galway)
  • Formatas: Hardback, 550 pages, aukštis x plotis x storis: 235x161x33 mm, weight: 1088 g
  • Išleidimo metai: 13-Aug-2019
  • Leidėjas: Oxford University Press
  • ISBN-10: 0198832974
  • ISBN-13: 9780198832973
Kitos knygos pagal šią temą:
Invitation to Computational HomotopyDidesnis paveikslėlis
  • Formatas: Hardback, 550 pages, aukštis x plotis x storis: 235x161x33 mm, weight: 1088 g
  • Išleidimo metai: 13-Aug-2019
  • Leidėjas: Oxford University Press
  • ISBN-10: 0198832974
  • ISBN-13: 9780198832973
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780198832973.html
Raktažodžiai:
An Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques.

An Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques.

The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues which will appeal to readers already familiar with basic theory and who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules.

These topics are covered in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include in-depth examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject.

An Invitation to Computational Homotopy serves as a self-contained and informal introduction to these topics and their implementation in the sphere of computer science. Written in a dynamic and engaging style, it skilfully showcases a range of useful machine computations, and will serve as an invaluable aid to graduate students working with algebraic topology.

Recenzijos

It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques. * Graham Ellis, Mathematical Reviews * The book is well equipped with exercises...providing a widespread of approaches to the material in the book.this book is definitely one I appreciate having read, and one that I can see myself recommending to students who already wish to go into something related to group theory. * Mikael Vejdemo-Johansson, Assistant Professor of Data Science at CUNY College of Staten Island, Mathematical Association of America *

List of Figures
xvii
List of Tables
xxi
List of Algorithms
xxiii
1 Path Components and the Fundamental Group
1(126)
1.1 Regular CW-spaces
2(12)
1.2 Simplicial, Cubical and Permutahedral Complexes
14(17)
1.3 Path Components and Persistence
31(15)
1.3.1 Mapper Clustering
42(4)
1.4 Simple Homotopy
46(12)
1.5 Non-regular CW-spaces
58(7)
1.6 The Fundamental Group
65(25)
1.6.1 Seifert-van Kampen Theorem for Groupoids
85(1)
1.6.2 The Wirtinger Presentation
86(4)
1.7 Computing with fp Groups
90(12)
1.8 Computing with fp Quandles
102(6)
1.9 Covering Spaces
108(11)
1.9.1 A Remark on Flat Manifolds
118(1)
1.10 Cayley Graphs and Presentations
119(4)
1.11 Exercises
123(4)
2 Cellular Homology
127(86)
2.1 Chain Complexes and Euler Integrals
128(9)
2.2 Euler Characteristics and Group Presentations
137(7)
2.3 Chain Maps and Homotopies
144(6)
2.4 Homology over Fields
150(15)
2.5 Homotopical Data Fitting
165(7)
2.6 Homology over Principal Ideal Domains
172(10)
2.7 Excision
182(5)
2.8 Cohomology Rings
187(21)
2.8.1 Van Kampen Diagrams and Cup Products
202(6)
2.9 Exercises
208(5)
3 Cohomology of Groups
213(114)
3.1 Basic Definitions and Examples
215(9)
3.2 Small Finite Groups
224(7)
3.3 Operations on Resolutions
231(18)
3.3.1 Perturbed Actions
246(3)
3.4 The Transfer Map
249(4)
3.5 Finite E-groups
253(13)
3.5.1 Modular Isomorphism Problem
260(6)
3.6 Lie Algebras
266(5)
3.7 Group Cohomology Rings
271(10)
3.8 Spectral Sequences
281(8)
3.9 A Test for Cohomology Ring Completion
289(7)
3.9.1 Computing Kernels of Derivations
295(1)
3.10 Cohomology Operations
296(15)
3.10.1 Stiefel-Whitney Classes
306(5)
3.11 Bredon Homology
311(7)
3.12 Coxeter Groups
318(5)
3.13 Exercises
323(4)
4 Cohomological Group Theory
327(52)
4.1 Standard Cocycles
328(4)
4.2 Classification of Group Extensions
332(5)
4.3 Coefficient Modules
337(2)
4.4 Crossed Modules
339(6)
4.5 A Five Term Exact Sequence
345(12)
4.6 The Nonabelian Tensor Product
357(11)
4.7 Crossed and Relative Group Extensions
368(3)
4.8 More on Relative Homology
371(2)
4.9 Exercises
373(6)
5 Cohomology of Homotopy 2-types
379(44)
5.1 Outline
379(3)
5.2 The Fundamental Crossed Module
382(9)
5.2.1 Maps from a Surface to the Projective Plane
388(3)
5.3 Finite Crossed Modules
391(11)
5.4 Simplicial Objects
402(11)
5.5 The Homological Perturbation Lemma
413(4)
5.6 Homology of Simplicial Groups
417(3)
5.7 Exercises
420(3)
6 Explicit Classifying Spaces
423(68)
6.1 Review of Constructions
424(2)
6.2 Aspherical Groups
426(5)
6.3 Graphs of Groups
431(10)
6.3.1 One-relator Groups
434(2)
6.3.2 The Group SL2(Z[ 1/m])
436(5)
6.4 Triangle Groups
441(12)
6.4.1 Cyclic Central Extensions of Triangle Groups
444(2)
6.4.2 Poincare's Theorem
446(4)
6.4.3 Generalized Triangle Groups
450(3)
6.5 Non-positive Curvature
453(3)
6.6 Coxeter Groups Revisited
456(10)
6.7 Artin Groups
466(14)
6.7.1 Some Cohomology Rings
476(4)
6.8 Arithmetic Groups
480(8)
6.9 Exercises
488(3)
Appendix
491(16)
A.1 Primer on Topology
491(2)
A.2 Primer on Category Theory
493(1)
A.3 Primer on Finitely Presented Groups and Groupoids
494(2)
A.4 Homology Software
496(1)
A.5 Software for Group Cohomology
497(1)
A.6 Parallel Computation
498(6)
A.7 Installing HAP and Related Software
504(3)
Bibliography 507(14)
Index 521
Graham Ellis received his PhD from the University of Wales, Bangor, in 1984 and has taught at the National University of Ireland, Galway since 1987. He is the Established Professor of Mathematics at Galway. He has published research articles on computational algebra, group theory and low-dimensional topology.