List of contributors 

xvii  
I PLENARY LECTURES 

3  (254) 

1 Roger PenroseA Personal Appreciation 


3  (6) 





1 Personal and historical remarks 


3  (1) 


4  (1) 

3 Integrable systems and solitons 


5  (1) 


5  (1) 


6  (1) 


6  (1) 


7  (2) 

2 Hypercomplex Manifolds and the Space of Framings 


9  (22) 






9  (1) 


10  (3) 


13  (1) 


14  (6) 

5 Twistor spaces and isomonodromic deformations 


20  (4) 

6 Holonomy and hypergeometric functions 


24  (5) 


29  (2) 

3 Gauge Theory in Higher Dimensions 


31  (18) 










31  (1) 


31  (2) 


33  (4) 


37  (1) 

5 The twodimensional picture 


38  (2) 

6 Adiabatic limits and dimension reduction 


40  (2) 

7 An example: quadrics in P(5) 


42  (1) 

8 Vanishing cycles and pseudoholomorphic curves 


43  (2) 


45  (2) 


47  (2) 

4 Noncommutative Differential Geometry and the Structure of SpaceTime 


49  (32) 




Foreword 

49  (1) 


49  (5) 


54  (10) 

3 The local index formula and the transverse fundamental class 


64  (5) 

4 The notion of manifold and the axioms of geometry 


69  (6) 

5 The spectral geometry of spacetime 


75  (3) 


78  (3) 

5 Einstein's Equation and Conformal Structure 


81  (18) 






81  (1) 

2 Asymptotic simplicity and conformal Einstein equations 


82  (2) 

3 De Sittertype spacetimes 


84  (1) 

4 Antide Sittertype spacetimes 


84  (2) 

5 Minkowskitype spacetimes 


86  (8) 

5.1 Conformal Minkowski space 


86  (1) 

5.2 Some existence results 


87  (1) 


88  (1) 

5.4 Assumptions on the data 


89  (1) 

5.5 Gauge conditions and conformal field equations 


90  (1) 

5.6 The finite regular initial value problem near spacelike infinity 


91  (1) 

5.7 The total characteristic at spacelike infinity 


92  (1) 

5.8 Comments on our procedure 


93  (1) 


94  (3) 


97  (2) 

6 Twistors, Geometry, and Integrable Systems 


99  (10) 






99  (1) 

2 Twistors for 3dimensional spacetime 


99  (2) 

3 An integrable YangMillsHiggs system 


101  (2) 


103  (1) 


104  (2) 


106  (1) 


107  (2) 

7 On FourDimensional Einstein Manifolds 


109  (14) 






109  (1) 

2 The curvature of 4manifolds 


110  (1) 

3 The HitchinThorpe inequality 


111  (2) 


113  (1) 

5 SeibergWitten techniques 


114  (3) 


117  (3) 


120  (1) 


120  (3) 

8 Loss of Information in Black Holes 


123  (12) 





1 Personal and historical remarks 


123  (1) 


124  (11) 

9 Fundamental Geometry: the PenroseHameroff `Orch OR' Model of Consciousness 


135  (26) 





1 Introduction: on the trail of an enigma 


135  (1) 

2 Philosophy: a panexperiential `fundamentality' 


135  (3) 

3 Physics: objective reduction (OR) 


138  (2) 

4 Biology: quantum coherence in microtubules? 


140  (10) 


140  (1) 

4.2 Frohlich's biological coherence 


141  (1) 

4.3 Quantum isolationavoiding environmental interaction and decoherence 


142  (2) 

4.4 Macroscopic quantum coherence and gap junctions 


144  (2) 

4.5 Evolution, Orch OR and the Cambrian explosion 


146  (4) 

5 Summary of the `Orch OR' model of consciousness 


150  (2) 

6 Assumptions and testable predictions of Orch OR 


152  (2) 

7 Conclusion: Penrose's Platonic world 


154  (1) 


155  (6) 

10 Implications of Transience for Spacetime Structure 


161  (12) 






170  (3) 

11 Geometric Issues in Quantum Gravity 


173  (22) 






173  (4) 


173  (2) 


175  (2) 


177  (5) 


177  (1) 

2.2 Quantum configuration space 


178  (2) 

2.3 Kinematical Hilbert space 


180  (2) 


182  (7) 


182  (1) 


183  (2) 


185  (2) 

3.4 Properties of area operators 


187  (2) 


189  (3) 


192  (1) 


192  (3) 

12 From Quantum Codemaking to Quantum Codebreaking 


195  (20) 





1 What is wrong with classical cryptography? 


195  (2) 

2 Is the Bell theorem of any practical use? 


197  (2) 

3 Quantum key distribution 


199  (2) 


201  (2) 

5 Public key cryptosystems 


203  (2) 

6 Fast and slow algorithms 


205  (1) 


206  (3) 


209  (2) 


211  (1) 


212  (3) 

13 Penrose Tilings and Quasicrystals Revisited 


215  (12) 






215  (3) 

2 New approach to Penrose tiling: single tile/matching rule 


218  (2) 

3 New approach to Penrose tiling: maximizing cluster density 


220  (3) 


223  (2) 


225  (2) 

14 Decaying Neutrinos and the Geometry of the Universe 


227  (8) 






227  (1) 

2 Relic neutrinos as dark matter 


228  (2) 

3 Decaying neutrinos and the ionisation of the universe 


230  (2) 

4 A new observational test of the decaying neutrino theory 


232  (1) 


233  (2) 

15 Quantum Geometric Origin of All Forces in String Theory 


235  (10) 






235  (1) 

2 Forces and local symmetries 


235  (1) 

3 Fieldtheoretic KaluzaKlein 


236  (2) 


238  (2) 

5 Stringtheoretic KaluzaKlein 


240  (1) 

6 Sduality and the big fix 


241  (2) 


243  (2) 

16 Space from the Point of View of Loop Groups 


245  (12) 





1 Loop groups and quantum field theory 


246  (3) 

2 Loop groups and lowdimensional topology 


249  (1) 


250  (2) 


252  (5) 
II PARALLEL SESSION I: QUANTUM THEORY AND BEYOND 

257  (50) 

17 The Twistor Diagram Programme 


257  (8) 






262  (3) 

18 Geometric Models for Quantum Statistical Inference 


265  (12) 










265  (1) 


266  (3) 


269  (1) 


270  (1) 

5 Quantum statistical estimation 


271  (1) 

6 Higher order variance bounds 


272  (2) 

7 Quantum geometry vs information geometry 


274  (1) 


274  (3) 

19 Spin Networks and Topology 


277  (14) 






277  (1) 

2 Networks and discrete space 


277  (1) 

3 The bracket state summation and the Jones polynomial 


278  (4) 

3.1 The Reidemeister moves 


279  (3) 


282  (5) 


287  (4) 

20 The Physics of Spin Networks 


291  (16) 






291  (1) 

2 Spin networks in nonperturbative quantum gravity 


292  (7) 


299  (2) 


301  (6) 
III PARALLEL SESSION II: GEOMETRY AND GRAVITY 

307  (42) 

21 The Sen Conjecture for Distinct Fundamental Monopoles 


307  (10) 






307  (1) 


308  (1) 

3 Weak XXX strong coupling 


308  (1) 

4 Bound states at threshold 


309  (1) 


309  (3) 


312  (1) 


313  (1) 


314  (1) 

9 Bound states in the continuum 


315  (1) 


315  (2) 

22 An Unorthodox View of GR via Characteristic Surfaces 


317  (8) 














317  (2) 

2 The null surface formulation of GR 


319  (3) 


319  (1) 

2.2 Imposing the Einstein equations 


319  (1) 

2.3 Selfdual Einstein equations 


320  (1) 

2.4 Asymptotically flat GR 


321  (1) 

3 Discussion and applications 


322  (1) 


323  (2) 

23 Amalgamated CodazziRaychaudhuri Identity for Foliation 


325  (12) 






325  (3) 


328  (2) 

3 The adapted foliation connection 


330  (1) 

4 The amalgamated foliation curvature tensor 


331  (3) 


334  (3) 

24 Abstract/Virtual/Reality/Complexity 


337  (12) 






337  (1) 


338  (2) 


340  (2) 


342  (1) 


343  (2) 


345  (4) 
IV PARALLEL SESSION III: FUNDAMENTAL QUESTIONS IN QUANTUM MECHANICS 

349  (34) 

25 InteractionFree Measurements 


349  (8) 





1 The Penrose bomb testing problem 


349  (2) 

2 The ElitzurVaidman bomb testing problem 


351  (1) 

3 Experimental realization of the IFM 


352  (1) 


353  (1) 

5 Applications of the IFM 


354  (1) 

6 The IFM as counterfactuals 


354  (1) 


355  (2) 

26 Quantum Measurement Problem and the Gravitational Field 


357  (12) 






357  (1) 

2 Quantum measurement problem 


358  (2) 

3 Efforts to resolve the measurement problem 


360  (3) 

4 Gravitational reduction of the wave packet 


363  (4) 


367  (2) 

27 Entanglement and Quantum Computation 


369  (14) 






369  (1) 

2 Quantum computation and complexity 


370  (3) 

3 Superposition and entanglement in quantum computation 


373  (3) 

4 Entanglement and the superfast quantum Fourier transform 


376  (1) 


377  (1) 


378  (5) 
V PARALLEL SESSION IV: MATHEMATICAL ASPECTS OF TWISTOR THEORY 

383  (40) 

28 Penrose Transform for Flag Domains 


383  (12) 





1 The Penrose transform in formulas 


383  (3) 


383  (1) 


384  (1) 

1.3 Inverse Penrose transform 


385  (1) 

1.4 Holomorphic cohomology 


385  (1) 

1.5 Boundary integral formula 


386  (1) 

2 Generalized Penrose transform (geometrical problems) 


386  (4) 

2.1 Holomorphic cohomology 


386  (1) 


387  (1) 


388  (1) 


389  (1) 

3 Generalized Penrose transform (analytic problems) 


390  (2) 


392  (3) 

29 Twistor Solution of the Holonomy Problem 


395  (8) 










395  (1) 


395  (2) 

3 Twistor theory of holonomy groups 


397  (4) 


401  (2) 

30 The Penrose Transform and Real Integral Geometry 


403  (8) 






403  (1) 


403  (1) 

3 The holomorphic Penrose transform 


404  (1) 

4 The connection with integral geometry 


405  (1) 

5 A new method in real integral geometry 


406  (3) 

5.1 Pullback from RP(3) to F 


406  (1) 

5.2 Pushdown from F to Gr(2)(R(4)) 


407  (1) 


408  (1) 


409  (2) 

31 Pythagorean Spinors and Penrose Twistors 


411  (12) 






411  (1) 


411  (2) 

3 Projective quadrics and twistors 


413  (5) 


414  (3) 


417  (1) 


418  (5) 
VI AFTERWORD 

423  


423  





1 Geometry, and the roots and aims of twistor theory 


423  (1) 

2 Towards a twistor description of Einsteinian physics 


424  (5) 

3 Further issues of physics and biology 


429  