Sophie Morier-Genoud: An overview of q-analogs of real numbers

Many classical sequences of integers have well established q-analogs. The most fundamental ones are certainly the q-integers and the q-binomial coefficients which both appear in various areas of mathematics and physics. With Valentin Ovsienko we recently suggested a definition for the q-rational numbers based on combinatorial properties and on continued fraction expansions. The definition of q-rationals naturally extends the one of q-integers and leads to a ratio of polynomials with positive integer coefficients. A surprising phenomenon of stabilization allows us to define q-irrational numbers as formal power series with integer coefficients. These q-numbers have been studied in connection with many fields such as combinatorics of posets, braid groups and knots invariants,  homological algebra, Markov-Hurwitz approximation, cluster algebras. In this talk I will give an overview of the subject. I will recall the basic definitions and properties and present new symmetries on the q-reals.