Combinatorics of Words and Number Theory

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One of the classical problems of mathematics is to find solutions to Diophantine Equations (named after Diophantus of Alexandria, a Greek mathematician of the third century AD). More precisely, only integral solutions are considered in this context (unlike e.g. x^2=2, whose solution is the square root of 2, a non-integer). One famous such equation is the equation x^2+y^2=z^2, which recalls the Pythagorean theorem, and whose infinitely many integral solutions (e.g. x=3,y=4,z=5) are all known. But an even more famous example is the equation x^n+y^n=z^n, appearing in the so-called "Fermat's last theorem". The fact that this equation has no solutions when the exponent n is at least equal to 3, is a result proved by Andrew Wiles in the 1990's. My program of research is based on a new original approach to the Markoff Diophantine equation x^2+y^2+z^2=3xyz. Markoff was a brilliant Russian number-theorist who studied this equation (when he was 20), around 1880. He became even more famous later, through the theory of Markov processes. He himself wrote his name with two 'f', in his first, French-written, articles. It turns out that one may completely solve the Markoff equation, using combinatorics on words, in a manner that sheds new light on many aspects of the question. Combinatorics on words is a rather new branch of mathematics, with links to automata theory, algebra and number theory, and also linguistics.  The link with the Markoff equation exploits a certain construction on words, called palindromisation: by means of a certain algorithm, devised by Aldo de Luca, each word is turned into a palindrome. This construction, after some matrix computations, gives all triples which are solutions of the Markoff equation. I recently obtained this new solution of the equation in collaboration with two student of mine (Abram, Lapointe).  My research program will explore many aspects of new links between combinatorics of words and number theory.