In this 8 hours course, designed for PhD students and researchers, but also accessible to Master’s students in Mathematics and Applied Mathematic, we will present the main definitions of the theory of bialgebras, with various examples, from algebraic groups, or from combinatorics, especially examples with two interacting coproducts, which will be called double bialgebras. We will explain how the "doubling alphabet" trick allows to construct such objects and apply this for the construction of the bialgebra of quasi-symmetric functions. We will finally apply all these results to construct a bialgebra based on packed matrices, introduced last year by Claudia Malvenuto, Frédéric Patras and myself. I. Algebras, coalgebras, and bialgebras Definitions and examples: polynomials, (proto)-algebraic monoids, tensor algebras, graphs... II Double bialgebras Definitions and examples: polynomials, graphs... Fundamental polynomial invariant The doubling alphabet trick and construction of the double bialgebra of quasi-symmetric functions III A 2-dimensional generalization of quasi-symmetric functions Packed matrices and their presentation through products of alphabets Morphism from quasi-symmetric functions to packed matrices Mer 11 febbraio 10h00-12h00 Aula L Ven 13 febbraio 10h00-12h00 Aula F Mer 18 febbraio 10h00-12h00 Aula F Ven 20 febbraio 10h00-12h00 Aula F