Combinatorics on words and digital geometry

我的主要研究领域是词的组合学，其中算法的观点在理论和实践方面都起着重要的作用。我的研究沿着以下几条轴线展开，其中单词的组合是核心： A.无限词的组合数学 B。数字几何和切片 C.形式语言与应用 A.在这方面，我主要关注回文复杂性（不同回文子字的数量）和子字或因子复杂性（不同因子的数量）。它被用来分类单词，特别是相应的态射的不动点。我还研究了一类特殊的无限的话（称为平滑）有关的指数轨迹的奥尔登堡，其中包括不动点的指数轨迹，因此Kolakoski字。 B。离散图形的轮廓由所谓的弗里曼链码编码，弗里曼链码是由数字平面（正方形网格）中的四个基本步骤组成的4个字母字母表上的一个单词。因此，词的组合学直接适用，并为开发新算法提供了有用的工具。回文和有限斯特米安词再次出现，因为它们分别编码对称和线段。林登因子分解也适用于分割。通过平移使平面平铺的多角形的例子是许多问题的来源，这些问题也与数论有关。 C.我研究的是枚举问题，经常是用有理或代数语言枚举的。我还考虑了组合对象的穷举和随机生成的问题。

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.