From Christoffel Words to Markoff Numbers

Christophe Reutenauer, Professor of Mathematics, Université du Québec à Montréal

Christophe Reutenauer was educated at École Normale Supérieure in Paris, and subsequently undertook his postgraduate studies at the Université de Paris 6 in 1977, before going on to complete his doctorate thesis at the same institution in 1980. He was a former researcher at the CNRS (Centre National de la Recherche Scientifique) in Paris, and LITP (Laboratoire d'Informatique Théorique et de Programmation) from 1976 to 1990. Reutenauer has been a professor at UQAM (Université du Québec à Montréal) since 1985, and was also a professor at the University of Strasbourg between 1999 and 2001. Since then, he has been an invited professor or researcher at several universities, including Saarbrücken, Darmstadt, Roma, Napoli, Palermo, UQAM, San Diego (UCSD), Strasbourg, Montpelier, Bordeaux, Paris-Est, Nice, and the Mittag-Leffler Institute. He was also the Canadian Research Chair for Algebra, Combinatorics and mathematical Informatics between 2001 and 2015.

Part I - The theory of Markoff1. Basics2. Words3. Markoff numbers4. The Markoff property5. Continued fractions6. Binary quadratic forms7. Lagrange number8. Words and quadratic numbers9. Lagrange numbers less than three10. Markoff's Theorem for approximations11. Markoff's theorem for quadratic forms12. Numerology13. Historical notesPart II - The Theory of Christoffel words14. Palindromes and periods15. Conjugates and factors16. Standard words17. Lyndon words and Christoffel words18. Sturmian words19. Stern-Brocot Tree20. Free group on two generators21. The commutator subgroup SL2(Z)'22. Solution to selected exercises