CONTENTS -- Preface -- PART 1 Geometry, Theoretical Physics and Cosmology -- CHAPTER 1 The Emergence of Algebraic Geometry in Contemporary Physics -- 1. Introduction -- 2. Algebraic Geometry in a Nutshell -- 3. Gauge Theories -- 4. String Theory -- References -- CHAPTER 2 Quantum Gravity and Quantum Geometry -- 1. Introduction -- 2. Glimpses of Quantum Gravity -- 3. Strings and Geometry -- Recommended Reading List -- CHAPTER 3 The de Sitter and Anti-de Sitter Universes -- 1. Introduction -- 2. A Visual Description of the de Sitter Manifolds -- 2.1. Curved spaces of constant curvature -- 2.2. The de Sitter universe -- 2.3. Anti-de Sitter -- 3. De Sitter -- 3.1. Coordinate systems -- 3.2. Boundary at infinity. Geodesics -- 4. De Sitter Quantum Field Theory -- 4.1. Plane waves -- 4.2. Two-point functions of the Klein-Gordon quantum field -- 4.3. Generalised free fields -- 5. Lifetime of a de Sitter Particle -- 5.1. Two properties that are crucial -- 5.2. The model -- 5.3. Decay 1κ → 2ν -- 6. Anti-de Sitter -- 6.1. Notations and geometry -- 6.2. Quantum field theory -- 7. Correspondence with Conformal Field Theories on C2,d à la Lüscher-Mack -- 8. Two-Point Functions -- 8.1. The analytic structure of two-point functions on the AdS spacetime -- 8.2. The simplest example revisited: Klein-Gordon fields in the AdS/CFT correspondence -- References -- CHAPTER 4 Geometry and Topology in Relativistic Cosmology -- Overview -- 1. The Four Scales of Geometry -- 2. Curvature vs. Topology -- 3. Basics of Topology -- 3.1. Simple vs. multiple connectedness -- 3.2. Fundamental domain and holonomy group -- 3.3. Universal covering -- 3.4. Spaceforms -- 4. Three-Dimensional Manifolds of Constant Curvature -- 4.1. Euclidean space forms -- 4.2. Spherical space forms -- 4.3. Hyperbolic space forms -- 5. Topology and Cosmology -- 6. The Drumhead Universe.

7. The Dodecahedral Universe -- References -- PART 2 The Problem of Space in Neurosciences -- CHAPTER 5 Space Coding in the Cerebral Cortex -- 1. Introduction -- 2. The Traditional Concept. Space is Coded in Oculocentric Coordinates -- 3. Coding of Peripersonal Space in the Parieto-Frontal Circuits for Reaching -- 4. Further Cortical Areas Involved in Space Coding -- 5. Lesions Data Confirm the Presence of Different Types of Space Coding -- 6. Conclusions -- References -- CHAPTER 6 Action and Space Representation -- 1. Introduction -- 2. Evidence for Discrete Representations of Space in Humans -- 3. Re-mapping of Space by Tool Use -- 4. Space Representation During Walking -- 5. Conclusions -- References -- CHAPTER 7 The Space Representations in the Brain -- Overview -- 1. Introduction -- 2. The Multisensory Bases of the Space Representation -- 3. Several Representations of the Space in the Human Brain -- 4. Multiple Representations of Peripersonal Space -- 5. Multisensory Representation of Peripersonal Space for Action -- 6. Conclusion -- Acknowledgements -- References -- CHAPTER 8 The Enactive Constitution of Space -- 1. Introduction -- 2. Peripersonal Space as Body-centred and Multisensory Space -- 3. The Role of Bodily Movements in Constituting Space -- 4. Near and Far: How Action Shapes Space -- 5. Concluding Remarks -- References -- PART 3 Geometrical Methods in the Biological Sciences -- CHAPTER 9 Causes and Symmetries in Natural Sciences: The Continuum and the Discrete in Mathematical Modelling -- 1. Introduction -- 2. Causal Structures and Symmetries, in Physics -- 2.1. Symmetries as starting point for intelligibility -- 2.2. Time and causality in physics -- 2.3. Symmetry breakings and fabrics of interaction -- Intermezzo. Remarks and Technical Commentaries -- 3. From the Continuum to the Discrete.

3.1. Computer science and the philosophy of arithmetics -- 3.2. Laplace, digital rounding and iteration -- 3.3. Iteration and prediction -- 3.4. Rules and the algorithm -- 4. Causalities in Biology -- 4.1. Basic representation -- 4.2. On contingent finality -- 4.3. 'Causal' dynamics: development, maturity, aging, death -- 4.4. Invariants of causal reduction in Biology -- 4.5. A few comments and comparisons with physics -- 5. Synthesis and Conclusion -- References -- CHAPTER 10 Topological Invariants of Geometrical Surfaces and the Protein Folding Problem -- 1. Introduction -- 2. Protein Folding Problem -- 3. Inverse Folding Problem -- 4. Molecular Dynamic Simulations -- 5. Topological Invariant Number and the Folding Process -- 6. Protein Folding Inhibitor and Non-Conventional Drug Design -- 7. Conclusion -- References -- CHAPTER 11 The Geometry of Dense Packing and Biological Structures -- Overview -- 1. Introduction -- 2. Geometry of the {3, 3, 5}-Polytope in S3 -- 2.1. The {3, 3, 5}-polytope -- 2.2. The Hopf fibration of S3 -- 3. The Geometry of Helices -- 3.1. The Boerdijk-Coxeter chain of tetrahedra -- 3.2. Discretising the fibration for the {3, 3, 5} polytope -- 3.3. The Coxeter helix -- 3.4. The α-helix: a disclinated Coxeter helix -- 3.5. Other helices in proteins -- 4. Proteins as Close Packing -- 4.1. Laguerre and Voronoi cells in proteins -- 4.2. The protein '3chy' as an example of Laguerre tessellation -- 4.3. Cell statistics -- 4.4. Proteins versus random close packed structure -- 5. Disclination Lines in Proteins -- 5.1. More on disclinations -- 5.2. Network of disclinations in proteins -- 6. Conclusions -- Acknowledgements -- References -- CHAPTER 12 When Topology and Biology Meet 'For Life': The Interactions Between Topological Forms and Biological Functions -- Overview.

1. Remarks on the Unlinking of DNA Molecule and the Chromosome Segregation in vivo -- 1.1. Topological operations and biological functions -- 1.2. Some useful topological notions -- 2. Topological and Dynamical Aspects of DNA Structure and the Spatial Organisation of the Chromosome -- 2.1. Geometry of the double-helix and conformational modifications of chromatin -- 3. The Topological Role of Topoisomerases -- 4. The Relationship between the Linking Number and Supercoiling of DNA Molecule -- 4.1. Topological complexity of DNA and its biological meaning -- 4.2. The structural flexibility of biomolecules. DNA compaction by successive order of coiling -- 5. More about Topoisomerases and their Mathematical Abilities and Biological Functions -- 6. Tangles, Knotting, and DNA Recombination: the Close Link between Topological 'Information' Acting on Supramolecular Forms and Biological Processes -- 7. Condensation of the Double-Helix Molecule into the Chromatin, and the Role of Supercoiling -- 8. Topological Models for Chromosome Compaction -- the Mathematical Concepts of 'Linking Number', 'Twist', 'Writhe', and their Biological Meaning -- 9. A Mathematical Model for Explaining the Folding of Chromatin Fibre During Interphase -- 10. Biological Justifications for the above model -- 11. Open Mathematical Questions, Biological Implications, and Some Suggestions for the Future Research -- 12. Conclusion -- References -- About the Contributors -- INDEX.