New Trends in Geometry: Their Role in the Natural and Life Sciences

CONTENTS......Page 6
Preface......Page 8
PART 1 Geometry, Theoretical Physics and Cosmology......Page 12
1. Introduction......Page 14
2. Algebraic Geometry in a Nutshell......Page 16
3. Gauge Theories......Page 17
4. String Theory......Page 21
References......Page 25
1. Introduction......Page 28
2. Glimpses of Quantum Gravity......Page 30
3. Strings and Geometry......Page 32
Recommended Reading List......Page 43
1. Introduction......Page 46
2.1. Curved spaces of constant curvature......Page 49
2.2. The de Sitter universe......Page 51
2.3. Anti-de Sitter......Page 52
3. De Sitter......Page 55
3.1. Coordinate systems......Page 56
3.2. Boundary at infinity. Geodesics......Page 60
4.1. Plane waves......Page 62
4.2. Two-point functions of the Klein–Gordon quantum field......Page 65
4.3. Generalised free fields......Page 66
5.1. Two properties that are crucial......Page 67
5.2. The model......Page 68
5.3. Decay 1κ → 2ν......Page 69
6.1. Notations and geometry......Page 72
6.2. Quantum field theory......Page 76
7. Correspondence with Conformal Field Theories on C2,d à la Lüscher–Mack......Page 77
8.1. The analytic structure of two-point functions on the AdS spacetime......Page 80
8.2. The simplest example revisited: Klein–Gordon fields in the AdS/CFT correspondence......Page 82
References......Page 88
1. The Four Scales of Geometry......Page 92
2. Curvature vs. Topology......Page 94
3.1. Simple vs. multiple connectedness......Page 97
3.2. Fundamental domain and holonomy group......Page 98
3.4. Spaceforms......Page 99
4. Three-Dimensional Manifolds of Constant Curvature......Page 100
4.1. Euclidean space forms......Page 101
4.2. Spherical space forms......Page 102
4.3. Hyperbolic space forms......Page 103
5. Topology and Cosmology......Page 104
6. The Drumhead Universe......Page 107
7. The Dodecahedral Universe......Page 110
References......Page 113
PART 2 The Problem of Space in Neurosciences......Page 116
1. Introduction......Page 118
2. The Traditional Concept. Space is Coded in Oculocentric Coordinates......Page 120
3. Coding of Peripersonal Space in the Parieto-Frontal Circuits for Reaching......Page 122
4. Further Cortical Areas Involved in Space Coding......Page 127
5. Lesions Data Confirm the Presence of Different Types of Space Coding......Page 129
References......Page 133
1. Introduction......Page 138
2. Evidence for Discrete Representations of Space in Humans......Page 139
3. Re-mapping of Space by Tool Use......Page 141
4. Space Representation During Walking......Page 143
5. Conclusions......Page 145
References......Page 146
1. Introduction......Page 148
2. The Multisensory Bases of the Space Representation......Page 150
3. Several Representations of the Space in the Human Brain......Page 155
4. Multiple Representations of Peripersonal Space......Page 159
5. Multisensory Representation of Peripersonal Space for Action......Page 160
References......Page 164
1. Introduction......Page 168
2. Peripersonal Space as Body-centred and Multisensory Space......Page 169
3. The Role of Bodily Movements in Constituting Space......Page 171
4. Near and Far: How Action Shapes Space......Page 174
5. Concluding Remarks......Page 177
References......Page 178
PART 3 Geometrical Methods in the Biological Sciences......Page 182
1. Introduction......Page 184
2. Causal Structures and Symmetries, in Physics......Page 185
2.2. Time and causality in physics......Page 189
2.3. Symmetry breakings and fabrics of interaction......Page 192
Intermezzo. Remarks and Technical Commentaries......Page 194
3. From the Continuum to the Discrete......Page 196
3.1. Computer science and the philosophy of arithmetics......Page 197
3.2. Laplace, digital rounding and iteration......Page 198
3.3. Iteration and prediction......Page 201
3.4. Rules and the algorithm......Page 203
4. Causalities in Biology......Page 208
4.1. Basic representation......Page 209
4.2. On contingent finality......Page 213
4.3. ‘Causal’ dynamics: development, maturity, aging, death......Page 214
4.4. Invariants of causal reduction in Biology......Page 215
4.5. A few comments and comparisons with physics......Page 216
5. Synthesis and Conclusion......Page 217
References......Page 219
1. Introduction......Page 222
3. Inverse Folding Problem......Page 223
4. Molecular Dynamic Simulations......Page 224
5. Topological Invariant Number and the Folding Process......Page 226
7. Conclusion......Page 229
References......Page 231
Overview......Page 232
1. Introduction......Page 233
2.2. The Hopf fibration of S3......Page 234
3.1. The Boerdijk–Coxeter chain of tetrahedra......Page 237
3.2. Discretising the fibration for the {3, 3, 5} polytope......Page 238
3.3. The Coxeter helix......Page 239
3.4. The α-helix: a disclinated Coxeter helix......Page 241
3.5. Other helices in proteins......Page 242
4.1. Laguerre and Voronoi cells in proteins......Page 243
4.3. Cell statistics......Page 244
4.4. Proteins versus random close packed structure......Page 246
5.2. Network of disclinations in proteins......Page 248
References......Page 250
Overview......Page 254
1. Remarks on the Unlinking of DNA Molecule and the Chromosome Segregation in vivo......Page 255
1.1. Topological operations and biological functions......Page 256
1.2. Some useful topological notions......Page 258
2. Topological and Dynamical Aspects of DNA Structure and the Spatial Organisation of the Chromosome......Page 259
2.1. Geometry of the double-helix and conformational modifications of chromatin......Page 261
4. The Relationship between the Linking Number and Supercoiling of DNA Molecule......Page 267
4.1. Topological complexity of DNA and its biological meaning......Page 271
4.2. The structural flexibility of biomolecules. DNA compaction by successive order of coiling......Page 272
5. More about Topoisomerases and their Mathematical Abilities and Biological Functions......Page 273
6. Tangles, Knotting, and DNA Recombination: the Close Link between Topological ‘Information’ Acting on Supramolecular Forms and Biological Processes......Page 274
7. Condensation of the Double-Helix Molecule into the Chromatin, and the Role of Supercoiling......Page 280
8. Topological Models for Chromosome Compaction; the Mathematical Concepts of ‘Linking Number’, ‘Twist’, ‘Writhe’, and their Biological Meaning......Page 283
9. A Mathematical Model for Explaining the Folding of Chromatin Fibre During Interphase......Page 294
10. Biological Justifications for the above model......Page 302
11. Open Mathematical Questions, Biological Implications, and Some Suggestions for the Future Research......Page 306
12. Conclusion......Page 308
References......Page 310
About the Contributors......Page 318
INDEX......Page 324