Modular deformation of rational numbers and plane geometry

par Perrine Jouteur (MPIS Leipzig)

jeudi 3 sept. 2026, 17:00 → 18:00 Europe/Paris

Amphi Jordan (Bat. Braconnier)

The q-analogue of an object is a family of objects indexed by a variable q, such that the specialization q->1 gives back the initial object. One of the oldest deformed objects in mathematics are q-integers, already used by Euler and Gauss to solve combinatorial problems. 
In this talk, I will present a generalization of these q-integers to real numbers, defined in 2020 by Morier-Genoud and Ovsienko. The theory of q-real numbers is based on a deformed action of the modular group PSL2(Z) on the real line. I will give a geometric interpretation of this q-action, understanding q-real numbers as geodesics of the hyperbolic plane. This point of view naturally leads to consider new geometric operations on q-rational numbers, looking at common tangent lines of the geodesics. We will see that these operations have an algebraic counterpart that we call Springborn operations, and that are quadratic versions of the Farey sum. I will describe a bridge between the geometric and the algebraic settings and highlight how one can use this bridge to deduce structural results on q-rational numbers. This talk is based on a joint work with Olga Paris-Romaskevich and Alexander Thomas.