Non[com,mut]ative geometry                                        &                            Supersymmetry              ...
Geometry & physics   Classical physics: flat space, timeThijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 3...
Geometry & physics   Classical physics: flat space, time   General relativity (Einstein, 1916):Thijs van den Broek (RU) | ...
Geometry & physics   Classical physics: flat space, time   General relativity (Einstein, 1916):                     Curved...
Geometry & physics   Classical physics: flat space, time   General relativity (Einstein, 1916):                     Curved...
Geometry & physics   Classical physics: flat space, time   General relativity (Einstein, 1916):                     Curved...
Geometry & physics   Classical physics: flat space, time   General relativity (Einstein, 1916):                     Curved...
?           Does this set-up allow for                      generalisations to                      noncommutative spacesW...
Noncommutative geometry | Basics      Analogy:      QM:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12   4Wednesday, Ma...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
Noncommutative geometry | Basics      Analogy:      QM:                                            acting on wave function...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Details       To be a bit more precise, we’re working with in the case       of GRT:                                ...
NCG | Action      Wish:               An actionThijs van den Broek (RU) | FOM Veldhoven | 17/1/12   6Wednesday, May 30, 2012
NCG | Action      Wish:               An action                                 Ac#on  /            Feynman      Cross    ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:               An action                          That is fixed by the noncommutative geometry     ...
NCG | Action      Wish:                An action                           That is fixed by the noncommutative geometry   ...
?           How different are the internal                      and external degrees of                      freedom of a ...
Standard Model | Set up              The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8Wednes...
Standard Model | Set up              The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8Wednes...
Standard Model | Set up              The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8Wednes...
Standard Model | Set up                                                            “Gauge group”              The geometry...
Standard Model | Set up                                                            “Gauge group”              The geometry...
Standard Model | Set up                                                            “Gauge group”              The geometry...
Standard Model | Set up                                                            “Gauge group”              The geometry...
Standard Model | Set up                                                            “Gauge group”              The geometry...
Standard Model | Predictions         Bonus: spectral action leads toThijs van den Broek (RU) | FOM Veldhoven | 17/1/12    ...
Standard Model | Predictions         Bonus: spectral action leads toThijs van den Broek (RU) | FOM Veldhoven | 17/1/12    ...
Standard Model | Predictions         Bonus: spectral action leads to                                                      ...
Standard Model | Predictions         Bonus: spectral action leads to                                                      ...
Standard Model | Predictions         Bonus: spectral action leads to                                                      ...
Standard Model | Predictions         Bonus: spectral action leads to                                                      ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
Supersymmetry & MSSM              N = 1 supersymmetry: each particle in the theory has a              superpartner, differ...
NCG & Supersymmetry  Complication:  Different origins of  particlesThijs van den Broek (RU) | FOM Veldhoven | 17/1/12   11...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                      SUSY   NCG  Complication:        ...
NCG & Supersymmetry                                                                         SUSY       NCG  Complication: ...
I Thank you.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12       12Wednesday, May 30, 2012
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Noncommutative geometry and supersymmetry

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Noncommutative geometry and supersymmetry

  1. 1. Non[com,mut]ative geometry & Supersymmetry yrtemmysrepuS Is | Thijs van den Broek From | Radboud University Nijmegen / Nikhef With | Walter van Suijlekom & Wim BeenakkerWednesday, May 30, 2012
  2. 2. Geometry & physics Classical physics: flat space, timeThijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  3. 3. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916):Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  4. 4. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetimeThijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  5. 5. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime MetricThijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  6. 6. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Metric Einstein equations:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  7. 7. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Metric Einstein equations: Or: Einstein - Hilbert actionThijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
  8. 8. ? Does this set-up allow for generalisations to noncommutative spacesWednesday, May 30, 2012 3
  9. 9. Noncommutative geometry | Basics Analogy: QM:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  10. 10. Noncommutative geometry | Basics Analogy: QM: acting on wave functions.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  11. 11. Noncommutative geometry | Basics Analogy: QM: acting on wave functions.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  12. 12. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  13. 13. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  14. 14. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Noncommutative space (...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  15. 15. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Noncommutative space (...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  16. 16. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Determines metric Noncommutative space (...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  17. 17. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  18. 18. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  19. 19. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...) GRT is a special example.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4Wednesday, May 30, 2012
  20. 20. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  21. 21. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  22. 22. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  23. 23. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  24. 24. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  25. 25. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  26. 26. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  27. 27. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  28. 28. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  29. 29. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5Wednesday, May 30, 2012
  30. 30. NCG | Action Wish: An actionThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  31. 31. NCG | Action Wish: An action Ac#on  /   Feynman   Cross Lagrangian rules sec#ons  Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  32. 32. NCG | Action Wish: An action That is fixed by the noncommutative geometry Ac#on  /   Feynman   Cross Lagrangian rules sec#ons  Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  33. 33. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons  Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  34. 34. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  35. 35. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  36. 36. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  37. 37. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  38. 38. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  39. 39. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  40. 40. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  41. 41. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: GRT: gives Einstein - Hilbert action + more ( )Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6Wednesday, May 30, 2012
  42. 42. ? How different are the internal and external degrees of freedom of a particle?Wednesday, May 30, 2012
  43. 43. Standard Model | Set up The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  44. 44. Standard Model | Set up The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  45. 45. Standard Model | Set up The geometry:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  46. 46. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  47. 47. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc).Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  48. 48. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  49. 49. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content. Spectral action gives: General Relativity + Standard Model + more ( )Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  50. 50. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content. Spectral action gives: General Relativity + Standard Model + more ( ) A geometrical understanding of the Standard Model!Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8Wednesday, May 30, 2012
  51. 51. Standard Model | Predictions Bonus: spectral action leads toThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  52. 52. Standard Model | Predictions Bonus: spectral action leads toThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  53. 53. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  54. 54. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constantsThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  55. 55. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constantsThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  56. 56. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constants Higgs  mass  predic#on:  158  -­‐  173  GeVThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9Wednesday, May 30, 2012
  57. 57. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  58. 58. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  59. 59. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  60. 60. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Gauge bosons (gluons, ...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  61. 61. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Sfermions (squarks, sleptons) Gauge bosons (gluons, ...) Gauginos (gluinos, ...)Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  62. 62. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Sfermions (squarks, sleptons) Gauge bosons (gluons, ...) Gauginos (gluinos, ...) Possible solution for various (theoretical) problems (e.g. provides dark matter candidate).Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10Wednesday, May 30, 2012
  63. 63. NCG & Supersymmetry Complication: Different origins of particlesThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  64. 64. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons GauginosThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  65. 65. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions:Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  66. 66. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric?Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  67. 67. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric?Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  68. 68. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   ✓ super-QCDThijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  69. 69. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   ✓ super-QCD General: soon to be finished.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  70. 70. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   Up next: ✓ super-QCD ? The Higgs mass. General: soon to be finished.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11Wednesday, May 30, 2012
  71. 71. I Thank you.Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 12Wednesday, May 30, 2012

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