Combinatorics of Words and Number Theory

数学的经典问题之一是找到丢番图方程的解（以公元三世纪的希腊数学家亚历山大的丢番图命名）。更准确地说，在这种情况下只考虑积分解（与例如x^2=2不同，其解是2的平方根，非整数）。其中一个著名的方程是方程x^2+y^2=z^2，它让人想起了毕达哥拉斯定理，它的无穷多个整数解（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“费马最后定理”中。当指数n至少等于3时，这个方程没有解的事实是Andrew Wiles在20世纪90年代证明的结果。我的研究计划是基于马尔可夫丢番图方程x^2+y^2+z^2=3xyz的一种新的原始方法。Markoff是一位杰出的俄罗斯数论家，他在1880年左右研究了这个方程（当时他20岁）。后来，他通过马尔可夫过程理论更加出名。他自己写了他的名字与两个'f'，在他的第一，法语书面，文章。事实证明，人们可以完全解决马尔可夫方程，使用组合数学的话，在某种程度上，揭示了新的光在许多方面的问题。词的组合学是一个相当新的数学分支，与自动机理论、代数和数论以及语言学都有联系。 与马尔可夫方程的联系利用了一种关于单词的特定结构，称为回文化：通过阿尔多·德·卢卡设计的某种算法，每个单词都变成了回文。这种结构，经过一些矩阵计算，给出了所有的三元组，这些三元组是马尔可夫方程的解。我最近与我的两个学生（Abram，Lapointe）合作获得了这个方程的新解。 我的研究计划将探索词的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.