Résumé : The goal of my lectures is to show what mathematics looks like from the univalent point of view. In the first lecture I will describe the foundational aspects of the univalent foundations, with first applications to finite sets, descent, and the construction of the projective spaces. In the second and third lectures I will describe how the univalence axiom shows up in group theory and in combinatorics.

Keywords : univalence axiom; homotopy type theory; univalent combinatorics; formalization of mathematics; projective spaces; group theory