Mathematics has been behind many of humanity's most significant advances in fields as varied as genome sequencing, medical science, space exploration, and computer technology. But those breakthroughs were yesterday. Where will mathematicians lead us tomorrow and can we help shape that destiny? This book assembles carefully selected articles highlighting and explaining cutting-edge research and scholarship in mathematics with an emphasis on three manifolds.
Author(s): Samuel F. Neilson
Year: 2006
Language: English
Pages: 172
NEW RESEARCH ON THREE-MANIFOLDS AND MATHEMATICS......Page 5
NOTICE TO THE READER......Page 6
CONTENTS......Page 7
PREFACE......Page 9
1 Introduction......Page 13
2 Notation......Page 16
3 Proof......Page 21
4 Remark to the Poincare Conjecture......Page 24
References......Page 25
Abstract......Page 29
2 A Quantum Gauge Model......Page 30
3 Classical Wilson Loop......Page 32
4 A Gauge Fixing Condition and Affine Kac-Moody Algebra......Page 34
5 Quantum Knizhnik-Zamolodchikov Equation in Dual Form......Page 39
6 Solving Quantum KZ Equation in Dual Form......Page 43
7 Computation of Quantum Wilson Lines......Page 44
8 Representing Braiding of Curves by Quantum Wilson Lines......Page 45
10 Defining Quantum Knots and Knot Invariant......Page 47
11 Examples of Quantum Knots......Page 48
12 Generalized Wilson Loops as Quantum Knots......Page 50
13 More Examples of Quantum Knots and Knot Invariant......Page 56
14 A Classification Table of Knots......Page 58
15 Examples of Quantum Links and Link Invariant......Page 74
16 Classification of Links......Page 76
17 Quantum Invariants of 3-manifolds and Classification......Page 80
18 An Investigation of the Proof of Poincare Conjecture......Page 90
19 Counterexamples of the Geometrization Conjecture......Page 93
References......Page 96
1 Introduction......Page 99
2 Proof of Theorem 1.1......Page 101
3 Additive Map Preserving Ranks and Geometry of Matrices......Page 105
References......Page 106
1 Preliminaries......Page 109
2 Motivation......Page 110
3 Exactness of the Expected Value......Page 111
4 Exactness of the Variance......Page 113
5 Error Bounds......Page 114
References......Page 117
1 Introduction......Page 119
2 Problem Formulation and the Scalings......Page 125
3 The Basic and Interaction Equations......Page 127
4 Ill-Posed System......Page 131
5 A Uniformly Valid Composite Approach......Page 132
6 Conclusion......Page 137
References......Page 139
1 Introduction......Page 143
2 Initial Layer Thickness......Page 145
3 Initial Layer Correction Equations......Page 148
4 Initial Layer Correction Functions......Page 149
5 Transcendental Smallness and Initial Layer Functions......Page 153
References......Page 155
1 Introduction......Page 157
2 Generalities......Page 158
3 Arithmetic Functions, Families of Rational Functions and Cardinality......Page 159
4 The Action of the Projective Line Homographies......Page 162
5 Constructions of Optical Orthogonal Codes (OOC)......Page 163
References......Page 167
1 Introduction......Page 169
2 L2-eigenforms of the Pauli-Dirac Operator......Page 171
3 L2-eigenprojector Kernels for the Pauli-Dirac Operator......Page 175
4 Resolvent Kernel for the Pauli-Dirac Operator......Page 176
5 Concluding Remarks......Page 177
References......Page 178
INDEX......Page 181
