Quantized rational numbers, from every angles

par Perrine Jouteur (Reims)

mercredi 30 avr. 2025, 14:00 → 15:00 Europe/Paris

112 (Bat. Braconnier)

Quantum analogues of real numbers are a generalization of q-deformed integers (also called Gaussian q-integers), which consist of replacing integers by polynomials in a formal variable "q", in such a way that the specialization q=1 is the initial number. This idea gives rise for example to generating series, and was already used by Euler to solve combinatorial problems. A good deformation must be compatible with the structural properties of the object that is being quantized. For instance, q-deformed binomial coefficients have a quantized Pascal rule.
In 2020, Sophie Morier-Genoud and Valentin Ovsienko proposed a quantization of rational numbers, generalizing the q-deformed integers, with good combinatorial properties. In this talk, we will define these q-rational numbers by three different (but equivalent) ways, first via continued fractions, then via the Farey graph, and finally via an action of the modular group. Then we will explain how this last definition discloses a new version of quantized rational numbers, known as the left one (in opposition to the previous q-deformed rational numbers, called the right ones), and we will extend the quantized action of the modular group to unify the left and the right versions.
If there is time left, we will have a glimpse on the relationship between quantized rational numbers and the braid theory, and how this relationship can suggest some multidimensional quantized rational numbers.