Combinatorics of Words and Number Theory

数学的经典问题之一是求出丢番图方程的解(以公元3世纪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整数解(不像x^2=2，其解是非整数2的平方根)。一个著名的这样的方程是方程x^2+y^2=z^2，它回忆起毕达哥拉斯定理，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，它出现在所谓的“费马大定理”中。当指数n至少等于3时，该方程无解的事实是由Andrew Wiles在1990年的《S》一文中证明的。 我的研究计划是基于马克夫丢番图方程x^2+y^2+z^2=3xyz的一种新的原始方法。马克夫是一位才华横溢的俄罗斯数论家，他在1880年左右研究了这个方程(当时他20岁)。后来，通过马尔可夫过程理论，他变得更加出名。在他的第一篇法语文章中，他自己用两个‘f’写了自己的名字。 事实证明，一个人可以完全解决马尔科夫方程，使用组合学的单词，以一种新的方式，揭示了问题的许多方面。 关于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。 与马尔科夫方程的联系利用了一种被称为回文的单词结构：通过阿尔多·德·卢卡设计的某种算法，每个单词都被变成了回文。这种构造，在一些矩阵计算之后，给出了所有的三元组，它们是马尔科夫方程的解。 最近，我和我的两个学生(艾布拉姆，拉波因特)合作，得到了这个方程的新解。 我的研究计划将探索单词组合学和数论之间许多新的联系。

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.