Amaury Freslon Université Paris-Saclay CRNS UMR 8628 Institut de Mathématique d'Orsay - Bâtiment 307 91405 Orsay Cedex France Courrier électronique : Bureau : 2L22 Bât. 307

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I am currently Maître de Conférence at Université Paris-Sud in the Topology and dynamics team. I am interested in the study of topological quantum groups, their structure and properties.

Here is my CV in english and in french.

Si vous cherchez ma page d'enseignement, elle se trouve ici.

Preprints

• Gaussian Generating functionals on easy quantum groups (with A. Skalski and U. Franz), Arxiv preprint pdf.
Abstract

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.

• Tracial central states on compact quantum groups (with A. Skalski and S. Wang), Arxiv preprint pdf.
Abstract

Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on q-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups.

Publications

1. Discrete quantum subgroups of free unitary quantum groups (with M. Weber), to appear in J. London Math. Soc. : pdf (old version).
Abstract

We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups U_N⁺. In other words, we classify all discrete quantum subgroups of Û_N⁺, thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.

2. Classical actions of quantum permutation groups (with F. Taipe and S. Wang), to appear in J. Operator Theory : link and pdf (old version).
Abstract

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions.

3. Advances on quantum permutation groups, in Advances in functional analysis and operator theory (M.V. Markin, I.V. Nikolaev and C. Trunk eds.), Contemp. Math. 798 (2024), 153-197 : link.
Abstract

We survey several results, ranging over the past fifteen years, concerning the quantum permutation groups and their quantum subgroups around three themes: quantum information, dynamics, and probability theory. In each case, we try to be as self-contained as possible and convey some intuition concerning the role and importance of quantum permutations, as well as key ideas of the proofs.

4. Tannaka-Krein reconstruction and ergodic actions of easy quantum groups (with F. Taipe and S. Wang) Comm. Math. Phys. 399 (2023), 105-172 : link and pdf (old version).
Abstract

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

5. The Gaussian part of a compact quantum group (with U. Franz and A. Skalski), J. Geom. Phys. 184 (2023), 104710 : link and pdf (old version).
Abstract

We introduce the Gaussian part of a compact quantum group G, namely the largest quantum subgroup of G supporting all the Gaussian functionals of G. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and q-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.

6. Free wreath products with amalgamation, Comm. Alg. 51 (2023), n^o 1, 72-94 : link and pdf (old version).
Abstract

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name "free wreath products of pairs" in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

7. Cutoff profiles for quantum Lévy processes and quantum random transpositions (with L. Teyssier and S. Wang), Probab. Theory Related Fields 183 (2022), 1285-1327: link and pdf (old version).
Abstract

We establish the existence of a cutoff phenomenon for a natural analogue of the Brownian motion on free orthogonal quantum groups. We compute in particular the cutoff profile, whose type is different from the previously known examples and involves free Poisson laws and the semi-circle distribution. We prove convergence in total variation (and even in L^p-norm for all p greater than 1) at times greater than the cutoff time and convergence in distribution for smaller times. We also study a similar process on quantum permutation groups, as well as the quantum random transposition walk. The latter yields in particular a quantum analogue of a recent result of the second-named author on random transpositions.

8. On the classification of partition quantum groups, Exp. Math. 39 (2021), n^o 2, 238-270 : link and pdf (old version).
Abstract

This is a survey on some results obtained recently in the classification of compact quantum groups associated to partitions, with a focus on the non-crossing case. We take a global look at the main results in the subject and highlight some key features of the methods used. We conclude by several suggestions for pushing further the classification.

9. Positive definite functions and cut-off for discrete groups, Canad. Math. Bull. 64 (2021), n^o 2, 306-322 : link and pdf.
Abstract

We consider the sequence of powers a fixed positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using surface groups and free groups again.

10. Topological generation and matrix models for quantum reflection groups (with M. Brannan and A. Chirvasitu), Adv. Math. 363 (2020) : link and pdf.
Abstract

We establish several new topological generation results for the quantum permutation groups \(S_N^+\) $S$_N⁺ and the quantum reflection groups $H$_N^s+. We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for $S$_N⁺ are inner faithful.

11. On two-coloured noncrossing partition quantum groups, Trans. Amer. Math. Soc. 372 (2019), n^o 6, 4471-4508 : link and pdf (old version).
Abstract

We classify compact quantum groups associated to noncrossing partitions coloured with two elements $x$ and $y$ which are their own inverses. Together with the work of P. Tarrago and M. Weber, this completes the classification of all noncrossing quantum groups on two colours. We also give some general structure results concerning noncrossing quantum groups and suggest some more general classification statements.

12. On the representation theory of some noncrossing partition quantum groups, Algebr. Represent. Theory 23 (2019), n^o 3, 483-492 : link and pdf (old version).
Abstract

We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint work with M. Weber.

13. Quantum reflections, random walks and cut-off, Internat. J. Math. 29 (2018), n^o 14, 1850101 : link and pdf (old version).
Abstract

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of such reflections. We also study random walks on quantum reflection groups and more generally free wreath products of finite group by quantum permutation groups.

14. Cut-off phenomenon for random walks on free orthogonal quantum groups, Probab. Theory Related Fields 174 (2019), n^o3-4, 731-760 : link and pdf (old version).
Abstract

We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the random rotation walk on this quantum group has a cut-off at $Nln(N)/2(1-cos(\theta ))$. This is the first result of this type for genuine compact quantum groups.

15. Torsion and K-theory for some free wreath products (with R. Martos), Int. Math. Res. Not. 2020 (2020), n^o 6, 1639-1670 : link and pdf (old version).
Abstract

We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if $G$ is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then $G\; \wr S$_N⁺ also satisfies the strong Baum-Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO$_q(3).

16. Modelling questions for quantum permutations (with T. Banica), Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), n^o 2, 1-26 : link and pdf (old version).
Abstract

Given a quantum permutation group $G\; \subset \; S$_N⁺, with orbits having the same size $K$, we construct a universal matrix model $\pi \; :\; C(G)\; \to \; M$_K(C(X)), having the property that the images of the standard coordinates $u$_ij ∊ C(G) are projections of rank $\le \; 1$. Our conjecture is that this model is inner faithful under suitable algebraic assumptions, and is in addition stationary under suitable analytic assumptions. We fully discuss this conjecture for classical groups and prove it for several families of duals of discrete groups.

17. The radial MASA in free orthogonal quantum groups (with R. Vergnioux), J. Funct. Anal. 271 (2016), n^o 10, 2776-2807: link and pdf (old version).
Abstract

We prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing, and we compute the associated bimodule. The proof relies on new properties of the Jones-Wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations.

18. Wreath products of finite groups by quantum groups (with A. Skalski), J. Noncommut. Geom. 12 (2018), n^o 1, 29-68 : link and pdf (old version).
Abstract

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon. We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.

19. On the partition approach to Schur-Weyl duality and free quantum groups (with an appendix by A. Chirvasitu), Transform. Groups 22 (2017), n^o 3, 707-751 : link and pdf (old version).
Abstract

We give a general definition of classical and quantum groups whose representation theory is "determined by partitions" and study their structure. This encompasses many examples of classical groups for which Schur-Weyl duality is described with diagram algebras as well as generalizations of P. Deligne's interpolated categories of representations. Our setting is inspired by many previous works on easy quantum groups and appears to be well-suited to the study of free fusion semirings. We classify free fusion semirings and prove that they can always be realized through our construction, thus solving several open questions. This suggests a general decomposition result for free quantum groups which in turn gives information on the compact groups whose Schur-Weyl duality is implemented by partitions. The paper also contains an appendix by A. Chirvasitu proving simplicity results for the reduced C*-algebras of some free quantum groups.

20. On bi-free De Finetti theorems (with M. Weber), Ann. Math. Blaise Pascal 23 (2016), n^o 1, 21-51 : link and pdf (old version).
Abstract

We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of $n$-freeness.

21. Permanence of approximation properties for discrete quantum groups, Ann. Inst. Fourier 65 (2015), n^o 4, 1437-1467 : link and pdf (old version).
Abstract

We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.

22. Fusion (semi)rings arising from quantum groups, J. Algebra 417 (2014), 161-197 : link and pdf (revised version).
Abstract

We study the fusion rings arising from easy quantum groups. We classify all the possible free ones, answering a question of T. Banica and R. Vergnioux. We then classify the possible groups of one-dimensional representations for free easy quantum groups. As an application, we give a unified proof of the Haagerup property for a broad class of easy quantum groups, recovering as special cases previous results by M. Brannan and F. Lemeux.

23. On the representation theory of partition (easy) quantum groups (with M. Weber), J. Reine Angew. Math. 720 (2016), 155-197 : link and pdf (old version).
Abstract

Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S.L. Woronowicz. In the case of easy quantum groups, the intertwiner spaces are given by the combinatorics of partitions, see the inital work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group $S$_N⁺, the free orthogonal quantum group $O$_N⁺ as well as the hyperoctahedral quantum group $H$_N⁺.

24. Graphs of quantum groups and K-amenability (with P. Fima), Adv. Math. 260 (2014), 233-280 : link and pdf (old version).
Abstract

Building on a construction of J-P. Serre, we associate to any graph of C*-algebras a maximal and a reduced fundamental C*-algebra and use this theory to construct the fundamental quantum group of a graph of discrete quantum groups. To illustrate the properties of this construction, we then prove that if all the vertex quantum groups are amenable, the fundamental quantum group is K-amenable. This generalizes previous results of Julg-Valette, R. Vergnioux and P. Fima.

25. CCAP for universal discrete quantum groups (with K. De Commer and M. Yamashita, with an appendix by S. Vaes), Comm. Math. Phys. 331 (2014), n^o 2, 677-701 : link and pdf (old version).
Abstract

In this paper we show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results also hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and $SU$_q(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for $SU$_q(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.

26. Examples of weakly amenable discrete quantum groups, J. Funct. Anal. 265 (2013), n^o 9, 2164-2187 : link and pdf (old version).
Abstract

In this paper we give a polynomial bound for the completely bounded norm of the projections on coefficients of a fixed irreducible representation in free orthogonal quantum groups. This enables us to compute their Cowling-Haagerup constant, which happens to be equal to $1$. We then use an argument of monoidal equivalence to extend our result to other free orthogonal quantum groups and quantum automorphism groups of finite-dimensional C*-algebras. This gives in particular non-unimodular examples of weakly amenable discrete quantum groups which are not amenable.

27. A note on weak amenability for free products of discrete quantum groups, C. R. Acad. Sci. Paris 350 (2012), n^o 7-8, 403-406 : link and pdf (old version).
Abstract

In this paper we prove that if two discrete quantum groups are weakly amenable with Cowling-Haagerup constant equal to $1$, then their free product is also weakly amenable with Cowling-Haagerup constant equal to $1$.

Book

• Compact matrix quantum groups and their combinatorics, LMS Student Texts in Mathematics 106, Cambridge University Press (2023) : link .

Other writings

Reports

1. On the classification of non-crossing partition quantum groups, Oberwolfach reports 45 (2019).
2. Cut-off for quantum random walks, Oberwolfach reports 22 (2018).

Talks

Below are several documents used as materials for research talks.

Notes in english

• Notes from a talk on The search of a quantum unitary Brownian motion.
• Notes from a lecture series on The classification of partition quantum groups.
• Notes from a talk on Classical spaces and quantum symmetries.
• Notes from a talk on Cut-off for quantum random walks.
• Notes from a talk on Matricial approximation of quantum automorphism groups of graphs.
• An exposition of the Ergodic theorem for compact groups.

Slides in english

• Slides from a talk on Easy quantum actions.
• Slides from a talk on Quantum groups in the heat.
• Slides from a talk on How to (badly) quantum shuffle cards.
• Slides from a talk on Diffusion of quantum orthogonal Brownian motion.

Notes in french

• Notes from a talk on Property (T) and expander graphs.
• Notes from a talk on Partition algebras and free quantum groups.
• Notes from a talk on Approximation properties for quantum groups.
• An introduction to the Elliott program.
• An introduction to C*-simplicity.
• An exposition of A. Connes' result on the fundamental group of a property (T) factor.

Dissertations

• Études des groupes quantiques libres : Analyse, Algèbre et Probabilités, Habilitation à diriger des recherches (Habilitation thesis) : pdf and slides.
• Approximation properties for discrete quantum groups (PhD thesis) : pdf.
Abstract

This dissertation is concerned with the notion of approximation property for discrete quantum groups and in particular weak amenability. Our goal is to apply techniques from geometric group theory to the study of quantum groups.

We first give a definition of weak amenability in the setting of discrete quantum groups and we develop some aspects of the general theory, inspired by the classical case. We particularly focus on the notion of Cowling-Haagerup constant. We also define a notion of relative amenability in this context which allows us to prove an additional stability result. Similar results are worked out for the Haagerup property. Eventually, we adress the question of free products of weakly amenable discrete quantum groups. Using the work of E. Ricard and X. Qu on Kintchine inequalities for free products, we prove that if two discrete quantum groups have Cowling-Haagerup constant equal to 1, their free product again has Cowling-Haagerup equal to 1.

Secondly, we give examples of weakly amenable discrete quantum groups. To do this, we use the recent work of M. Brannan on the Haagerup property for free quantum groups together with ideas from various works on Haagerup inequalities. More precisely, we give a polynomial bound for the norm of projections on coefficients of an irreducible representation of a free orthogonal quantum groups which allows us to "cut off" M. Brannan's functions and compute the Cowling-Haagerup constant. Finally, we apply techniques of monoidal equivalence to extend these results to other classes of discrete quantum groups, some of which are not unimodular.

Miscellaneous

1. Mathématiques, exercices incontournables - MPSI/MP2I (3^ème edition) (avec M. Bages, P. Bernard, J. Freslon, M. H&eactute;zard, C. Lagrange, J. Levie, M. Mansuy et J. Poineau), Dunod (2023) : lien.
2. Mathématiques, exercices incontournables - PCSI/PTSI (3^ème edition) (avec M. Bages, P. Bernard, J. Freslon, M. H&eactue;zard, C. Lagrange, J. Levie, M. Mansuy et J. Poineau), Dunod (2023) : lien.

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Département de Mathématiques, Université Paris-Sud, Bât. 425, F-91405 Orsay Cedex ,France