Résumé : The theoretical setting of his talk is based essentially on the paper - http://hal.archives-ouvertes.fr/hal-00193180/ - published in a workshop about Combinatorial Physics [1]. This setting was used to prove, with Christophe Reutenauer, a conjecture by Alain Connes [2]. We give new insights and applications of this concept.
[1] G.H.E. Duchamp and C. Tollu, Sweedler’s duals and Schützenberger’s calculus, In K. Ebrahimi-Fard, M. Marcolli and W. van Suijlekom (eds), Combinatorics and Physics, p. 67–78, Amer. Math. Soc. (Contemporary Mathematics, vol. 539), 2011.
[2] G. Duchamp, C. Reutenauer, Un critère de rationalité provenant de la géométrie non-commutative (à la mémoire de Schützenberger), Inventiones Mathematicae, 128, 613-622, (1997).

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