Liberation theory for noncommutative homogeneous spaces

We discuss the liberation question, in the homogeneous space setting. Our first series of results concerns the axiomatization and classification of the families of compact quantum groups $G=(G_N)$ which are "uniform", in a suitable sense. We study then the quotient spaces of type $X=(G_M\times G_N)/(G_L\times G_{M-L}\times G_{N-L})$, and the liberation operation for them, with a number of algebraic and probabilistic results.Comment: 24 page

Associative Yang-Baxter equation and Fukaya categories of square-tiled surfaces

We show that all strongly non-degenerate trigonometric solutions of the associative Yang-Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya category of square-tiled surfaces. Along the way, we give a classification result for cyclic $A_\infty$-algebra structures on a certain Frobenius algebra associated with a pair of 1-spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf. arXiv:1601.06141), we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of 1-spherical twists and their inverses.Comment: 37 pages, 9 figures. Minor revision after a referee's comments. To appear in Advances in Mathematic

Natural Communication

In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science

Spectral transfer morphisms for unipotent affine Hecke algebras

In this paper we will give a complete classification of the spectral transfer morphisms between the unipotent affine Hecke algebras of the various inner forms of a given quasi-split absolutely simple algebraic group, defined over a non-archimidean local field $\textbf{k}$ and split over an unramified extension of $\textbf{k}$. As an application of these results, the results of [O4] on the spectral correspondences associated with such morphisms and some results of Ciubotaru, Kato and Kato [CKK] we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on the formal degrees and adjoint gamma factors for all unipotent discrete series characters of unramified simple groups of adjoint type defined over $\bf{k}$.Comment: 61 pages; We explained the comparison with Lusztig's parameterization of unipotent representations in more detai

Symétries de jauge non-commutative et diffusions pseudo-unitaires

This thesis is devoted to the study of two quite different questions, which are related by the tools that we used to study them.The first question is that of the definition of lattice gauge theories with a non-commutative structure group. Here, by non-commutative, we do not mean non-Abelian, but instead non-commutative in the general sense of non-commutative geometry.The second question is that of the behaviour of Brownian diffusions on non-compact matrix groups of a specific kind, namely groups of pseudo-orthogonal, pseudo-unitary or pseudo-symplectic matrices.In the first chapter, we investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a Zhang algebra. Zhang algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups.We are interested in non-commutative analogues of random gauge fields, which we describe through the random holonomy that they induce.We propose a general definition of a holonomy field with Zhang gauge symmetry, and construct such a field starting from a quantum Lévy process on a Zhang algebra.As an application, we define higher dimensional generalizations of the so-called master field.In the second chapter, we study matricial approximations of higher dimensional master fields constructed in the previous chapter.These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in the algebras of real, complex or quaternionic numbers) and letting the dimension of these blocks tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks.In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described.In the last two chapters, we use tools introduced (Zhang algebras and coloured Brauer diagrams) in the first two ones to study Brownian motion on pseudo-unitary matrices in high dimensions.We prove convergence in non-commutative distribution of the pseudo-unitary Brownian motions we consider, to free with amalgamation semi-groups under the hypothesis of convergence of the normalized signature of the metric.In the split case, meaning that at least asymptotically the metric has as much negative directions as positive ones, the limiting distribution is that of a free L\'evy process, which is a solution of a free stochastic differential equation.We leave open the question of such a realization of the limiting distribution in the general case.In addition, we provide (intriguing) numerical evidences for the convergence of the spectral distribution of such random matrices and make two conjectures. At the end of the thesis, we prove asymptotic normality for the fluctuations.Cette thèse est consacrée à l'étude de deux questions très différentes, reliées par les outils que nous utilisons pour les étudier. La première question est celle de la définition des théories de jauge sur un réseau avec un groupe de structure non commutatif. Ici, non commutatif ne signifie pas non Abelian, mais plutôt non commutatif au sens général de la géométrie non commutative. La deuxième question est celle du comportement des diffusions Browniennes sur des groupes matriciels non compacts d'un type spécifique, à savoir des groupes de matrices pseudo-orthogonales, pseudo-unitaires ou pseudo-symplectiques.Dans le premier chapitre, nous étudions des théories de jauge quantiques sur un réseau et leur limite continue sur le plan euclidien ayant une algèbre de Zhang pour groupe de stucture. Les algèbres de Zhang sont des analogues non commutatifs des groupes et contiennent la classe des groupes duaux de Voiculescu. Nous nous intéressons donc aux analogues non commutatifs des champs de jauges quantiques, que nous décrivons par l'holonomie aléatoire qu'ils induisent. Nous proposons une définition générale d'un champ d'holonomies ayant une symétrie de jauge présentant la structure d'une algèbre de Zhang, et construisons un tel champ à partir d'un processus quantique de Lévy sur une algèbre de Zhang.Dans le deuxième chapitre, nous étudions les approximations matricielles des champs maîtres en dimensions supérieures construits dans le chapitre précédent. Ces approximations (en distribution non commutative) sont obtenues en extrayant des blocs d'une diffusion unitaire Brownienne (à coefficients dans les algèbres de nombres réels, complexes ou quaternioniques) et en laissant la dimension de ces blocs tendre vers l'infini. Nous divisons notre étude en deux parties : dans la première, nous extrayons des blocs carrés tandis que dans la seconde, nous autorisons des blocs rectangulaires.Dans les deux derniers chapitres, nous utilisons les outils introduits (algèbres de Zhang et diagrammes de Brauer colorés) dans les deux premiers pour étudier des diffusions sur des groupes de matrices pseudo-unitaires. Nous prouvons la convergence non commutative des mouvements Browniens pseudo-unitaires que nous considérons vers des semi-groupes libres avec amalgamation sous l'hypothèse de convergence de la signature normalisée de la métrique de l'espace sous-jacent.Dans le cas déployé, c'est-à-dire, qu'au moins asymptotiquement, la métrique a autant de directions négatives que de directions positives, la distribution limite est la distribution d'un processus de Lévy, solution d'une équation différentielle stochastique libre. Nous laissons ouverte la question d'une telle réalisation de la distribution limite dans le cas général.De plus, nous présentons des résultats numériques sur la convergence de la distribution spectrale de ces matrices aléatoires et faisons deux conjectures. Dans le dernier chapitre, nous prouvons la normalité asymptotique des fluctuations

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems

Matricial approximations of higher dimensional master fields

We study matricial approximations of master fields constructed in [6]. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in R, C or K) and letting the dimension of these blocks to tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described

A walk in the noncommutative garden

This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the ``Tehran program'') of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.Comment: 106 pages, LaTeX, 23 figure

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast