Combinatorics on words and digital geometry

我的主要研究领域是词语组合学，其中算法的观点在理论和实践方面都起着重要的作用。我的研究方向以单词组合为中心：A.无限单词的组合B.数字几何和拼贴C.形式语言和应用A.在这个领域，我主要关注回文复杂性（不同回文子词的数量）和子词或因子复杂性（不同因子的数量）。它用于对单词进行分类，并相应地对语素的固定点进行分类。我还研究了一类与Oldenburger的指数轨迹相关的无限词（称为光滑），其中包括指数轨迹的不动点，因此也包括Kolakoski词。B.离散图形的轮廓由所谓的弗里曼链码编码，这是一个由数字平面（方形网格）中的四个基本步骤组成的4个字母字母表上的单词。因此，词的组合学可以直接应用，并为开发新的算法提供了有用的工具。回文和有限斯图尔曼字再次出现，因为它们分别编码对称和线段。林登分解也适用于分割。通过平移来平铺平面的多项式是许多问题的根源，这些问题也与数论有关。C.我研究枚举问题，通常用理性或代数语言来枚举。我还考虑了穷举和随机生成组合对象的问题。

My main research domain is combinatorics on words, where the algorithmic point of view plays an important role both on the theoretical and practical aspects. I developed my research along the following axes where combinatorics of words are central: A. Combinatorics of infinite words B. Digital geometry and tilings C. Formal languages and applications A. In this area I am mostly concerned with palindromic complexity (number of distinct palindomic subwords) and subword or factor complexity (number of distinct factors). It is used for classifying words and particularly fixed points of morphisms accordingly. I also study a particular class of infinite words (called smooth) related to the exponential trajectories of Oldenburger, which include the fixed points of exponential trajectories and therefore the Kolakoski word. B. The contour of discrete figures is encoded by the so-called Freeman chain code, a word on a 4-letter alphabet consisting of the four elementary steps in the digital plane (square grid). Combinatorics on words apply therefore directly and provide useful tools for developing new algorithms. Palindromes and finite Sturmian words again appear, since they encode respectively symmetries and line segments. Lyndon factorizations also apply in the segmentation. The case of polyominoes that tile the plane by translation is a source of numerous problems also linked with number theory. C. I study enumeration problems, often are enumerated by rational or algebraic languages. I also consider the problems of exhaustive and random generation of combinatorial objects.