Seminario: "Combinatorics on words for Markoff numbers" - Laurent Vuillon
Martedì 28 marzo 2023 - ore 15:30
Aula Seminari - Dipartimento di Scienze Matematiche
Politecnico di Torino
"Combinatorics on words for Markoff numbers"
Laurent Vuillon
Université de Savoie
Logic, Computer Science and Discrete Mathematics team
https://www.lama.univ-savoie.fr/pagesmembres/vuillon/indexang.html
Abstract: In a first part, we introduce the Markoff numbers which are fascinating integers related to number theory, Diophantine equation, hyperbolic geometry, continued fractions and Christoffel words. Many great mathematicians have worked on these numbers and the 100 years uniqueness conjecture by Frobenius is still unsolved. We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v in {a, b}* such that m - 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This work gives a recursive construction of Markoff numbers by the lengths of words generated by a discrete dynamical system.
In a second part, we show new results on Markoff numbers by using generalizations of Farey’s fractions and Christoffel words. In particular, we study a valuation of paths on the N^2 grid related to Markoff numbers. We show two conjectures of the famous book of Aigner by preserving the monotonicity of this valuation on a sequence of paths and by investigating the dynamic of local path transformations.
Video dell'evento: La registrazione del seminario è disponibile sul Canale YouTube del gruppo CrypTO.
