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$q$-analogues of numbers and of matrices.

Sophie Morier-Genoud (Université Reims Champagne Ardenne)
Mathematics Seminar
Wednesday, 2 March, 2022 | 14:15–15:00 | Zoom

Classical sequences of numbers often lead to interesting $q$-analogues. The most popular among them are certainly the $q$-integers and the $q$-binomial coefficients which both appear in various areas of mathematics and physics. It seems that $q$-analogues of rational numbers have been much less popular so far. With Valentin Ovsienko we recently suggested a notion of $q$-rationals based on combinatorial properties. The definition of $q$-rationals naturally extends the one of $q$-integers and leads to a ratio of polynomials with positive integer coefficients. I will explain the construction and give the main properties. I will mention connections with the combinatorics of posets, cluster algebras, Jones polynomials. Finally I will also present further developments of the theory, in particular I will focus on the notion of $q$-irrationals and $q$-unimodular matrices.

Organised by: AarHomAlg