q-analogues of real numbers

Abstract: The most popular q-analogues of numbers are certainly the q-integers and the q-binomial coefficients of Gauss which both appear in various areas of mathematics and physics. Most classical sequences of integers often have interesting q-analogues. With Valentin Ovsienko we recently suggested a notion of q-analogues for rational numbers. Our approach is based on combinatorial properties and continued fraction expansions of the rationals. The definition of q-rationals naturally extends the one of q-integers and leads to ratios 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. I will explain all the constructions and give the main properties of these q-numbers. The subject can be developed in connections with various topics such as the enumerative combinatorics, cluster algebras, homological algebra, Burau representation, Jones polynomials... I will briefly discuss some of these connections.

representation theory

Audience: researchers in the topic

RA Seminar

Series comments: The event format will include an online 50-minute lecture that will be held on the second Wednesday of every month. For the Zoom links and passwords, please subscribe to the mailing list (link and password will be emailed shortly before each talk).