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Knots and Primes: An Introduction to Arithmetic Topology 2012 [Minkštas viršelis]

3.33/5 (6 ratings by Goodreads)
  • Formatas: Paperback / softback, 191 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 42 Illustrations, black and white; XI, 191 p. 42 illus., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 29-Nov-2011
  • Leidėjas: Springer London Ltd
  • ISBN-10: 1447121570
  • ISBN-13: 9781447121572
Kitos knygos pagal šią temą:
Knots and Primes: An Introduction to Arithmetic Topology 2012Didesnis paveikslėlis
  • Formatas: Paperback / softback, 191 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 42 Illustrations, black and white; XI, 191 p. 42 illus., 1 Paperback / softback
  • Serija: Universitext
  • Išleidimo metai: 29-Nov-2011
  • Leidėjas: Springer London Ltd
  • ISBN-10: 1447121570
  • ISBN-13: 9781447121572
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781447121572.html
Raktažodžiai:
This book offers a foundation for arithmetic topology, a new branch of mathematics focused upon the analogy between knot theory and number theory. The coverage includes background information and theory, along with numerous useful examples and illustrations.

This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory.    Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Common aspects of both knot theory and number theory, for instance knots in three manifolds versus primes in a number field, are compared throughout the book. These comparisons begin at an elementary level, slowly building up to advanced theories in later chapters. Definitions are carefully formulated and proofs are largely self-contained.   When necessary, background information is provided and theory is accompanied  with a number of useful examples and illustrations, making this a useful text for both undergraduates and graduates in the field of knot theory, number theory and geometry. ?

Recenzijos

This is one of the best textbook I have seen in the last few years. this books is amazing! I really enjoyed it and I hope you will also enjoy it. It definitely should be part of your library if you work in number theory and/or topology. This book will become a classical very soon! (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, June, 2016)

The book under review is the first systematic treatment of the subject in a format suitable for a textbook. The book is rich in material for anybody interested in either the arithmetic or the topological side, and the connections and interactions are presented in a very convincing and detailed way. (Matilde Marcolli, Mathematical Reviews, March, 2013)

Once youve lived long enough in mathematics, the themes addressed in Knots and Primes: An Introduction to Arithmetic Topology are both familiar and exceedingly attractive. This is a fascinating topic and Morishitas book is an important contribution. it will spur a lot of work in this beatiful and accessible area of contemporary mathematics. (Michael Berg, The Mathematical Association of America, May, 2012)

1 Introduction
1(8)
1.1 Two Ways that Branched out from C.F. Gauss---Quadratic Residues and Linking Numbers
1(3)
1.2 Geometrization of Number Theory
4(1)
1.3 The Outline of This Book
5(4)
2 Preliminaries---Fundamental Groups and Galois Groups
9(40)
2.1 The Case of Topological Spaces
9(15)
2.2 The Case of Arithmetic Rings
24(15)
2.3 Class Field Theory
39(10)
3 Knots and Primes, 3-Manifolds and Number Rings
49(6)
4 Linking Numbers and Legendre Symbols
55(6)
4.1 Linking Numbers
55(2)
4.2 Legendre Symbols
57(4)
5 Decompositions of Knots and Primes
61(8)
5.1 Decomposition of a Knot
61(3)
5.2 Decomposition of a Prime
64(5)
6 Homology Groups and Ideal Class Groups I---Genus Theory
69(8)
6.1 Homology Groups and Ideal Class Groups
69(1)
6.2 Genus Theory for a Link
70(3)
6.3 Genus Theory for Prime Numbers
73(4)
7 Link Groups and Galois Groups with Restricted Ramification
77(8)
7.1 Link Groups
77(3)
7.2 Pro-l Galois Groups with Restricted Ramification
80(5)
8 Milnor Invariants and Multiple Residue Symbols
85(26)
8.1 Fox Free Differential Calculus
85(8)
8.2 Milnor Invariants
93(6)
8.3 Pro-l Fox Free Differential Calculus
99(3)
8.4 Multiple Residue Symbols
102(9)
9 Alexander Modules and Iwasawa Modules
111(14)
9.1 Differential Modules
111(5)
9.2 The Crowell Exact Sequence
116(4)
9.3 Complete Differential Modules
120(2)
9.4 The Complete Crowell Exact Sequence
122(3)
10 Homology Groups and Ideal Class Groups II---Higher Order Genus Theory
125(16)
10.1 The Universal Linking Matrix for a Link
125(3)
10.2 Higher Order Genus Theory for a Link
128(4)
10.3 The Universal Linking Matrix for Primes
132(2)
10.4 Higher Order Genus Theory for Primes
134(7)
11 Homology Groups and Ideal Class Groups III---Asymptotic Formulas
141(10)
11.1 The Alexander Polynomial and Homology Groups
141(3)
11.2 The Iwasawa Polynomial and p-Ideal Class Groups
144(7)
12 Torsions and the Iwasawa Main Conjecture
151(10)
12.1 Torsions and Zeta Functions
151(5)
12.2 The Iwasawa Main Conjecture
156(5)
13 Moduli Spaces of Representations of Knot and Prime Groups
161(10)
13.1 Character Varieties of Complex Representations of a Knot Group
161(1)
13.2 The Character Variety of Complex 1-Dimensional Representations of a Knot Group and Alexander Ideals
162(2)
13.3 Universal Deformation Spaces of p-Adic Representations of a Prime Group
164(1)
13.4 The Universal Deformation Space of p-Adic 1-Dimensional Representations of a Prime Group and Iwasawa Ideals
165(6)
14 Deformations of Hyperbolic Structures and p-Adic Ordinary Modular Forms
171(10)
14.1 Deformation of Hyperbolic Structures
171(3)
14.2 Deformation of p-Adic Ordinary Modular Galois Representations
174(7)
References 181(8)
Index 189