Combinatorics of Words and Number Theory

丢番图方程（以公元三世纪希腊数学家亚历山大的丢番图命名）的解是数学中的一个经典问题。更准确地说，在这种情况下只考虑积分解（不像x^2=2，其解是2的平方根，一个非整数）。其中一个著名的方程是方程x^2+y^2=z^2，它让人想起毕达哥拉斯定理，它的无穷多个积分解（例如x=3,y=4,z=5）都是已知的。但一个更著名的例子是方程x^n+y^n=z^n，出现在所谓的“费马大定理”中。当n的指数至少等于3时，这个方程没有解，这是安德鲁·怀尔斯在20世纪90年代证明的结果。我的研究项目是基于马尔科夫丢芬图方程x^2+y^2+z^2=3xyz的一种新的原始方法。马尔科夫是一位杰出的俄罗斯数论学家，他在1880年左右研究了这个方程（当时他20岁）。后来，通过马尔可夫过程理论，他变得更加出名。在他最初用法语写的文章中，他自己的名字中有两个“f”。事实证明，一个人可以完全解决马尔科夫方程，使用单词组合学，以一种对问题的许多方面都有新的认识的方式。单词组合学是数学的一个相当新的分支，它与自动机理论、代数和数论以及语言学都有联系。与马尔科夫方程的联系利用了单词的某种结构，称为回文化：通过阿尔多·德·卢卡设计的某种算法，每个单词都变成了一个回文。这种构造，经过一些矩阵计算，给出了作为马尔科夫方程解的所有三元组。我最近与我的两个学生（亚伯兰，拉普特）合作得到了这个方程的新解。我的研究计划将探索词语组合学和数论之间的新联系的许多方面。

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.