Teo Banica

Professeur de Mathematiques
Universite de Cergy-Pontoise

Bureau: St-Martin E-521
Tel: +331 34 25 66 80

Research

I'm interested in mathematical physics, and more specifically in math input for quantum mechanics, and for astrophysics.
I've been working on this since 1994, and wrote about 100 papers. Here is my CV, and profile at Google Scholar.

Books

I am currently writing several books, mostly of graduate level, and of quantum flavor, as follows:
  1. Introduction to quantum groups
  2. Invitation to Hadamard matrices
  3. Affine noncommutative geometry
  4. Quantum permutation groups
  5. Classical groups and Haar integration
  6. Methods of classical and free probability
  7. Quantum symmetries and lattice models
There will be some further books, also of basic level, on quantum mechanics and related topics.

Advice

If you want to learn some math and physics, you should go with the classics. Nothing beats the good old books from the 60s-70s, completed with a few more recent books, from the 80s-90s. Here are my book recommendations.

Regarding more recent math and physics books, including mine, these are quite often rather experimental, based on math and physics beliefs that are not proved yet to be useful. Buy or download, and read, at your own risk.

Videos

Here is a series of slide books, on topics similar to those of my main books, with video presentations:
  1. Introduction to linear algebra (2020)
  2. Introduction to matrix groups (2020)
  3. Introduction to operator algebras (2020)
  4. Introduction to quantum groups (2020)
  5. Introduction to noncommutative geometry (2020)
  6. Introduction to subfactor theory (2020)
  7. Introduction to free probability (2020)
  8. Introduction to Hadamard matrices (2020)

Papers

And here are my scientific papers, mostly in quantum algebra, with the surveys highlighted:
  1. On the polar decomposition of circular variables, Integral Equations Operator Theory 24 (1996), 372-377.
  2. The representation theory of free orthogonal quantum groups, C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), 241-244.
  3. The free unitary compact quantum group, Comm. Math. Phys. 190 (1997), 143-172.
  4. Hopf algebras and subfactors associated to vertex models, J. Funct. Anal. 159 (1998), 243-266.
  5. Representations of compact quantum groups and subfactors, J. Reine Angew. Math. 509 (1999), 167-198.
  6. Fusion rules for representations of compact quantum groups, Exposition. Math. 17 (1999), 313-337.
  7. Symmetries of a generic coaction, Math. Ann. 314 (1999), 763-780.
  8. Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000), 80-99.
  9. Subfactors associated to compact Kac algebras, Integral Equations Operator Theory 39 (2001), 1-14.
  10. Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys. 226 (2002), 221-232.

  11. The planar algebra of a coaction, J. Operator Theory 53 (2005), 119-158.
  12. Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (2005), 27-51.
  13. Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243-280.
  14. with S. Moroianu, On the structure of quantum permutation groups, Proc. Amer. Math. Soc. 135 (2007), 21-29.
  15. with J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), 381-414.
  16. with B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), 277-302.
  17. with J. Bichon, Quantum automorphism groups of vertex-transitive graphs of order ≤ 11, J. Algebraic Combin. 26 (2007), 83-105.
  18. with D. Bisch, Spectral measures of small index principal graphs, Comm. Math. Phys. 269 (2007), 259-281.
  19. with J. Bichon and G. Chenevier, Graphs having no quantum symmetry, Ann. Inst. Fourier 57 (2007), 955-971.
  20. with B. Collins, Integration over quantum permutation groups, J. Funct. Anal. 242 (2007), 641-657.

  21. with R. Nicoara, Quantum groups and Hadamard matrices, Panamer. Math. J. 17 (2007), 1-24.
  22. with J. Bichon and B. Collins, Quantum permutation groups: a survey, Banach Center Publ. 78 (2007), 13-34.
  23. with J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384.
  24. with B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008), 942-961.
  25. A note on free quantum groups, Ann. Math. Blaise Pascal 15 (2008), 135-146.
  26. with R. Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 321-340.
  27. with J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74-114.
  28. Cyclotomic expansion of exceptional spectral measures, Internat. J. Math. 20 (2009), 275-297.
  29. with R. Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), 327-359.
  30. with R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501.

  31. with B. Collins and P. Zinn-Justin, Spectral analysis of the free orthogonal matrix, Int. Math. Res. Not. 17 (2009), 3286-3309.
  32. with J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864-2910.
  33. with J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010), 677-703.
  34. with B. Collins and J.-M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839-856.
  35. with R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137-2164.
  36. with D. Goswami, Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010), 343-356.
  37. with S. Curran and R. Speicher, Classification results for easy quantum groups, Pacific J. Math. 247 (2010), 1-26.
  38. The orthogonal Weingarten formula in compact form, Lett. Math. Phys. 91 (2010), 105-118.
  39. with S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1-14.
  40. with S.T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), 3-37.

  41. with S. Curran and R. Speicher, Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149 (2011), 435-462.
  42. with B. Collins and J.-M. Schlenker, On polynomial integrals over the orthogonal group, J. Combin. Theory Ser. A 118 (2011), 778-795.
  43. with J. Bichon and S. Curran, Quantum automorphisms of twisted group algebras and free hypergeometric laws, Proc. Amer. Math. Soc. 139 (2011), 3961-3971.
  44. with A. Skalski, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260 (2011), 3252-3282.
  45. with J.-M. Schlenker, Combinatorial aspects of orthogonal group integrals, Internat. J. Math. 22 (2011), 1611-1646.
  46. with S. Curran and R. Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. 40 (2012), 401-435.
  47. with A. Skalski, Quantum isometry groups of duals of free powers of cyclic groups, Int. Math. Res. Not. 9 (2012), 2094-2122.
  48. with J. Bichon and S. Natale, Finite quantum groups and quantum permutation groups, Adv. Math. 229 (2012), 3320-3338.
  49. Quantum permutations, Hadamard matrices, and the search for matrix models, Banach Center Publ. 98 (2012), 11-42.
  50. with A. Skalski and P.M. Soltan, Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys. 62 (2012), 1451-1466.

  51. with U. Franz and A. Skalski, Idempotent states and the inner linearity property, Bull. Pol. Acad. Sci. Math. 60 (2012), 123-132.
  52. with J. Bhowmick and K. De Commer, Quantum isometries and group dual subgroups, Ann. Math. Blaise Pascal 19 (2012), 17-43.
  53. with I. Nechita and K. Zyczkowski, Almost Hadamard matrices: general theory and examples, Open Syst. Inf. Dyn. 19 (2012), 1-26.
  54. with I. Nechita, Asymptotic eigenvalue distributions of block-transposed Wishart matrices, J. Theoret. Probab. 26 (2013), 855-869.
  55. with J. Bichon, B. Collins and S. Curran, A maximality result for orthogonal quantum groups, Comm. Algebra 41 (2013), 656-665.
  56. with A. Skalski, Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations, Proc. Lond. Math. Soc. 106 (2013), 980-1004.
  57. The defect of generalized Fourier matrices, Linear Algebra Appl. 438 (2013), 3667-3688.
  58. with I. Nechita, Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math. 161 (2013), 2367-2379.
  59. with I. Nechita and J.-M. Schlenker, Analytic aspects of the circulant Hadamard conjecture, Ann. Math. Blaise Pascal 21 (2014), 25-59.
  60. First order deformations of the Fourier matrix, J. Math. Phys. 55 (2014), 1-22.

  61. with I. Nechita and J.-M. Schlenker, Submatrices of Hadamard matrices: complementation results, Electron. J. Linear Algebra 27 (2014), 197-212.
  62. Counting results for thin Butson matrices, Electron. J. Combin. 21 (2014), 1-14.
  63. Truncation and duality results for Hopf image algebras, Bull. Pol. Acad. Sci. Math. 62 (2014), 161-179.
  64. with I. Nechita, Block-modified Wishart matrices and free Poisson laws, Houston J. Math. 41 (2015), 113-134.
  65. with A. Skalski, The quantum algebra of partial Hadamard matrices, Linear Algebra Appl. 469 (2015), 364-380.
  66. with J. Bichon, Random walk questions for linear quantum groups, Int. Math. Res. Not. 24 (2015), 13406-13436.
  67. The glow of Fourier matrices: universality and fluctuations, Oper. Matrices 9 (2015), 457-474.
  68. Liberations and twists of real and complex spheres, J. Geom. Phys. 96 (2015), 1-25.
  69. Quantum isometries of noncommutative polygonal spheres, Munster J. Math. 8 (2015), 253-284.
  70. with S. Meszaros, Uniqueness results for noncommutative spheres and projective spaces, Illinois J. Math. 59 (2015), 219-233.

  71. The algebraic structure of quantum partial isometries, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), 1-36.
  72. A duality principle for noncommutative cubes and spheres, J. Noncommut. Geom. 10 (2016), 1043-1081.
  73. Half-liberated manifolds, and their quantum isometries, Glasg. Math. J. 59 (2017), 463-492.
  74. Liberation theory for noncommutative homogeneous spaces, Ann. Fac. Sci. Toulouse Math. 26 (2017), 127-156.
  75. Quantum isometries, noncommutative spheres, and related integrals, Banach Center Publ. 111 (2017), 101-144.
  76. with I. Nechita, Flat matrix models for quantum permutation groups, Adv. Appl. Math. 83 (2017), 24-46.
  77. with I. Patri, Maximal torus theory for compact quantum groups, Illinois J. Math. 61 (2017), 151-170.
  78. Deformed Fourier models with formal parameters, Studia Math. 239 (2017), 201-224.
  79. Quantum groups from stationary matrix models, Colloq. Math. 148 (2017), 247-267.
  80. Weingarten integration over noncommutative homogeneous spaces, Ann. Math. Blaise Pascal 24 (2017), 195-224.

  81. with J. Bichon, Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017), 519-540.
  82. with J. Bichon, Complex analogues of the half-classical geometry, Munster J. Math. 10 (2017), 457-483.
  83. with A. Chirvasitu, Thoma type results for discrete quantum groups, Internat. J. Math. 28 (2017), 1-23.
  84. The planar algebra of a fixed point subfactor, Ann. Math. Blaise Pascal 25 (2018), 247-264.
  85. with I. Nechita, Almost Hadamard matrices with complex entries, Adv. Oper. Theory 3 (2018), 149-189.
  86. with A. Freslon, Modelling questions for quantum permutations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), 1-26.
  87. Complex Hadamard matrices with noncommutative entries, Ann. Funct. Anal. 9 (2018), 354-368.
  88. Super-easy quantum groups: definition and examples, Bull. Pol. Acad. Sci. Math. 66 (2018), 57-68.
  89. with D. Ozteke and L. Pittau, Isolated partial Hadamard matrices, and related topics, Open Syst. Inf. Dyn. 25 (2018), 1-27.
  90. Tannakian duality for affine homogeneous spaces, Canad. Math. Bull. 61 (2018), 483-494.

  91. Unitary easy quantum groups: geometric aspects, J. Geom. Phys. 126 (2018), 127-147.
  92. with A. Chirvasitu, Quasi-flat representations of uniform groups and quantum groups, J. Algebra Appl. 18 (2019), 1-27.
  93. Higher transitive quantum groups: theory and models, Colloq. Math. 156 (2019), 1-14.
  94. Quantum groups, from a functional analysis perspective, Adv. Oper. Theory 4 (2019), 164-196.
  95. Higher orbitals of quizzy quantum group actions, Adv. Appl. Math. 109 (2019), 1-37.
  96. Quantum groups under very strong axioms, Bull. Pol. Acad. Sci. Math. 67 (2019), 83-99.
  97. Block-modified Wishart matrices: the easy case, Indiana Univ. Math. J. 69 (2020), 1-34.
  98. Homogeneous quantum groups and their easiness level, Kyoto J. Math. 61 (2021), 171-205.
  99. with J.P. McCarthy, The Frucht property in the quantum group setting, Glasg. Math. J., to appear.

Contact

teo.banica@gmail.com