- Quantum isometries and noncommutative geometry, in preparation.

The free complex sphere S^{N-1}_{\mathbb C,+} is the noncommutative manifold defined by the equations \sum_ix_ix_i^*=\sum_ix_i^*x_i=1. Certain submanifolds X\subset S^{N-1}_{\mathbb C,+}, related to the quantum groups, are known to have Riemannian features, including an integration functional. We review here the known facts on the subject.

- Complex Hadamard matrices and applications, book 2019.

A complex Hadamard matrix is a square matrix H\in M_N(\mathbb C) whose entries are on the unit circle, |H_{ij}|=1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F_N=(w^{ij}) with w=e^{2\pi i/N}. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects.

- Free quantum groups and related topics, book 2019.

The unitary group U_N has a free analogue U_N^+, and the study of the closed subgroups G\subset U_N^+ is a problem of general interest. We review here the general theory of U_N^+ and its subgroups, with all the needed preliminaries included. We discuss as well a number of more advanced topics, selected for their beauty, and potential importance.

- with A. Chirvasitu, Modelling questions for transitive quantum groups, preprint 2017.

- Homogeneous quantum groups and their easiness level, Kyoto J. Math., to appear.
- Block-modified Wishart matrices: the easy case, Indiana Univ. Math. J., to appear.

- Quantum groups under very strong axioms, Bull. Pol. Acad. Sci. Math. 67 (2019), 83-99.
- Higher orbitals of quizzy quantum group actions, Adv. Appl. Math. 109 (2019), 1-37.
- Quantum groups, from a functional analysis perspective, Adv. Oper. Theory 4 (2019), 164-196.
- Higher transitive quantum groups: theory and models, Colloq. Math. 156 (2019), 1-14.
- with A. Chirvasitu, Quasi-flat representations of uniform groups and quantum groups, J. Algebra Appl. 18 (2019), 1-27.

- Unitary easy quantum groups: geometric aspects, J. Geom. Phys. 126 (2018), 127-147.
- Tannakian duality for affine homogeneous spaces, Canad. Math. Bull. 61 (2018), 483-494.
- with D. Ozteke and L. Pittau, Isolated partial Hadamard matrices, and related topics, Open Syst. Inf. Dyn. 25 (2018), 1-27.
- Super-easy quantum groups: definition and examples, Bull. Pol. Acad. Sci. Math. 66 (2018), 57-68.
- Complex Hadamard matrices with noncommutative entries, Ann. Funct. Anal. 9 (2018), 354-368.
- with A. Freslon, Modelling questions for quantum permutations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), 1-26.
- with I. Nechita, Almost Hadamard matrices with complex entries, Adv. Oper. Theory 3 (2018), 149-189.
- The planar algebra of a fixed point subfactor, Ann. Math. Blaise Pascal 25 (2018), 247-264.

- with A. Chirvasitu, Thoma type results for discrete quantum groups, Internat. J. Math. 28 (2017), 1-23.
- with J. Bichon, Complex analogues of the half-classical geometry, Munster J. Math. 10 (2017), 457-483.
- with J. Bichon, Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017), 519-540.
- Weingarten integration over noncommutative homogeneous spaces, Ann. Math. Blaise Pascal 24 (2017), 195-224.
- Quantum groups from stationary matrix models, Colloq. Math. 148 (2017), 247-267.
- Deformed Fourier models with formal parameters, Studia Math. 239 (2017), 201-224.
- with I. Patri, Maximal torus theory for compact quantum groups, Illinois J. Math. 61 (2017), 151-170.
- with I. Nechita, Flat matrix models for quantum permutation groups, Adv. Appl. Math. 83 (2017), 24-46.
- Quantum isometries, noncommutative spheres, and related integrals, Banach Center Publ. 111 (2017), 101-144.
- Liberation theory for noncommutative homogeneous spaces, Ann. Fac. Sci. Toulouse Math. 26 (2017), 127-156.
- Half-liberated manifolds, and their quantum isometries, Glasg. Math. J. 59 (2017), 463-492.

- A duality principle for noncommutative cubes and spheres, J. Noncommut. Geom. 10 (2016), 1043-1081.
- The algebraic structure of quantum partial isometries, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), 1-36.

- with S. Meszaros, Uniqueness results for noncommutative spheres and projective spaces, Illinois J. Math. 59 (2015), 219-233.
- Quantum isometries of noncommutative polygonal spheres, Munster J. Math. 8 (2015), 253-284.
- Liberations and twists of real and complex spheres, J. Geom. Phys. 96 (2015), 1-25.
- The glow of Fourier matrices: universality and fluctuations, Oper. Matrices 9 (2015), 457-474.
- with J. Bichon, Random walk questions for linear quantum groups, Int. Math. Res. Not. 24 (2015), 13406-13436.
- with A. Skalski, The quantum algebra of partial Hadamard matrices, Linear Algebra Appl. 469 (2015), 364-380.
- with I. Nechita, Block-modified Wishart matrices and free Poisson laws, Houston J. Math. 41 (2015), 113-134.

- Truncation and duality results for Hopf image algebras, Bull. Pol. Acad. Sci. Math. 62 (2014), 161-179.
- Counting results for thin Butson matrices, Electron. J. Combin. 21 (2014), 1-14.
- with I. Nechita and J.-M. Schlenker, Submatrices of Hadamard matrices: complementation results, Electron. J. Linear Algebra 27 (2014), 197-212.
- First order deformations of the Fourier matrix, J. Math. Phys. 55 (2014), 1-22.
- with I. Nechita and J.-M. Schlenker, Analytic aspects of the circulant Hadamard conjecture, Ann. Math. Blaise Pascal 21 (2014), 25-59.

- with I. Nechita, Almost Hadamard matrices: the case of arbitrary exponents, Discrete Appl. Math. 161 (2013), 2367-2379.
- The defect of generalized Fourier matrices, Linear Algebra Appl. 438 (2013), 3667-3688.
- with A. Skalski, Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations, Proc. Lond. Math. Soc. 106 (2013), 980-1004.
- with J. Bichon, B. Collins and S. Curran, A maximality result for orthogonal quantum groups, Comm. Algebra 41 (2013), 656-665.
- with I. Nechita, Asymptotic eigenvalue distributions of block-transposed Wishart matrices, J. Theoret. Probab. 26 (2013), 855-869.

- with I. Nechita and K. Zyczkowski, Almost Hadamard matrices: general theory and examples, Open Syst. Inf. Dyn. 19 (2012), 1-26.
- with J. Bhowmick and K. De Commer, Quantum isometries and group dual subgroups, Ann. Math. Blaise Pascal 19 (2012), 17-43.
- with U. Franz and A. Skalski, Idempotent states and the inner linearity property, Bull. Pol. Acad. Sci. Math. 60 (2012), 123-132.
- with A. Skalski and P.M. Soltan, Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys. 62 (2012), 1451-1466.
- Quantum permutations, Hadamard matrices, and the search for matrix models, Banach Center Publ. 98 (2012), 11-42.
- with J. Bichon and S. Natale, Finite quantum groups and quantum permutation groups, Adv. Math. 229 (2012), 3320-3338.
- with A. Skalski, Quantum isometry groups of duals of free powers of cyclic groups, Int. Math. Res. Not. 9 (2012), 2094-2122.
- with S. Curran and R. Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. 40 (2012), 401-435.

- with J.-M. Schlenker, Combinatorial aspects of orthogonal group integrals, Internat. J. Math. 22 (2011), 1611-1646.
- with A. Skalski, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260 (2011), 3252-3282.
- with J. Bichon and S. Curran, Quantum automorphisms of twisted group algebras and free hypergeometric laws, Proc. Amer. Math. Soc. 139 (2011), 3961-3971.
- with B. Collins and J.-M. Schlenker, On polynomial integrals over the orthogonal group, J. Combin. Theory Ser. A 118 (2011), 778-795.
- with S. Curran and R. Speicher, Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149 (2011), 435-462.
- with S.T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), 3-37.

- with S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1-14.
- The orthogonal Weingarten formula in compact form, Lett. Math. Phys. 91 (2010), 105-118.
- with S. Curran and R. Speicher, Classification results for easy quantum groups, Pacific J. Math. 247 (2010), 1-26.
- with D. Goswami, Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010), 343-356.
- with R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137-2164.
- with B. Collins and J.-M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839-856.
- with J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010), 677-703.

- with J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864-2910.
- with B. Collins and P. Zinn-Justin, Spectral analysis of the free orthogonal matrix, Int. Math. Res. Not. 17 (2009), 3286-3309.
- with R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501.
- with R. Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), 327-359.
- Cyclotomic expansion of exceptional spectral measures, Internat. J. Math. 20 (2009), 275-297.
- with J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74-114.
- with R. Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 321-340.

- A note on free quantum groups, Ann. Math. Blaise Pascal 15 (2008), 135-146.
- with B. Collins, Integration over the Pauli quantum group, J. Geom. Phys. 58 (2008), 942-961.

- with J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384.
- with J. Bichon and B. Collins, Quantum permutation groups: a survey, Banach Center Publ. 78 (2007), 13-34.
- with R. Nicoara, Quantum groups and Hadamard matrices, Panamer. Math. J. 17 (2007), 1-24.
- with B. Collins, Integration over quantum permutation groups, J. Funct. Anal. 242 (2007), 641-657.
- with J. Bichon and G. Chenevier, Graphs having no quantum symmetry, Ann. Inst. Fourier 57 (2007), 955-971.
- with D. Bisch, Spectral measures of small index principal graphs, Comm. Math. Phys. 269 (2007), 259-281.
- with J. Bichon, Quantum automorphism groups of vertex-transitive graphs of order ≤ 11, J. Algebraic Combin. 26 (2007), 83-105.
- with B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), 277-302.
- with J. Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), 381-414.
- with S. Moroianu, On the structure of quantum permutation groups, Proc. Amer. Math. Soc. 135 (2007), 21-29.

- Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243-280.
- Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (2005), 27-51.
- The planar algebra of a coaction, J. Operator Theory 53 (2005), 119-158.

- Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys. 226 (2002), 221-232.

- Subfactors associated to compact Kac algebras, Integral Equations Operator Theory 39 (2001), 1-14.

- Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000), 80-99.

- Symmetries of a generic coaction, Math. Ann. 314 (1999), 763-780.
- Fusion rules for representations of compact quantum groups, Exposition. Math. 17 (1999), 313-337.
- Representations of compact quantum groups and subfactors, J. Reine Angew. Math. 509 (1999), 167-198.

- Hopf algebras and subfactors associated to vertex models, J. Funct. Anal. 159 (1998), 243-266.

- The free unitary compact quantum group, Comm. Math. Phys. 190 (1997), 143-172.

- The representation theory of free orthogonal quantum groups, C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), 241-244.
- On the polar decomposition of circular variables, Integral Equations Operator Theory 24 (1996), 372-377.