Une triangulation est un modèle de surface discrète consistant à recoller un nombre fini de triangles le long de leurs côtés de manière à obtenir une surface. Dans le cas planaire (i.e. quand la surface doit avoir la topologie de la sphère), l’étude de triangulations aléatoires repose sur des résultats de comptage exacts obtenus par Tutte dans les années 60.

En revanche, pour des surfaces plus générales, de tels résultats exacts ne sont plus disponibles. Le but de l’exposé sera de donner une idée des techniques développées pour étudier des triangulations aléatoires de grand genre malgré le manque de résultats combinatoires précis. On verra en particulier que parfois, les résultats probabilistes permettent même
d’obtenir en retour des estimées combinatoires. Travaux en commun avec Baptiste Louf et Guillaume Chapuy.

Many real-world optimization challenges are significantly harder than the scenarios that can be rigorously analyzed by mathematical means. In addition, practitioners often lack time and resources to develop problem-specific solution strategies; they therefore need to rely on general-purpose optimization heuristics such as local search variants, evolutionary and genetic algorithms, Bayesian optimization techniques, and similar. Unfortunately, practitioners often choose and apply these algorithms without much guidance from the scientific community.
With this presentation, we will discuss how (theoretical and empirical) research can be used to develop more principled usage of these randomized optimization heuristics.

Les séries génératrices comptant des marches dans le quadrant, les distributions stationnaires de marches aléatoires ou encore les transformées de Laplace de mouvements browniens réfléchis satisfont le même type d’équations fonctionnelles en deux variables catalytiques. Dans cet exposé, je présenterais des méthodes issues de la théorie de Galois classique ou différentielle pour déterminer le comportement algébrique de solutions de telles équations voire d’expliciter ces solutions en terme de fonctions spéciales.

Balanced binary search trees are a common data structure for implementing ordered sets, with three kinds of queries: checking whether a given value belongs to the set, inserting a value, and deleting a value. The two latter queries require updating the data structure.

Some implementations, like red-black trees or weak AVL trees, allow efficient update procedures, which run in a top-down way and require an amortized constant number of write operations per update query. Yet, AVL trees still require bottom-up updating procedures, which may require a logarithmic number of write operations per query. We will present new algorithms for updating AVL trees, directly inspired from algorithms for weak AVL trees, that bridge this gap: they run in a top-down way and/or require only an amortized constant number of write operations per update query. Most of the presentation will be devoted to the latter aspect.

Abstract:
The guess-and-prove paradigm often proves useful in combinatorics, when studying the nature of generating functions. The guessing phase relies on computations that « discover » linear dependencies between a generating function and its successive powers or its successive derivatives. This enables to formulate conjectures about the generating function, for example about its algebraicity (root of a polynomial) or D-finiteness (solution of a linear differential equation). The purpose of this talk is to discuss what happens behind the scenes when this type of guessing approach is used. The key actors are different flavours of the so-called « Padé approximants » and « rational interpolants », which are small-degree algebraic relations between the above-mentioned powers or derivatives. Discovering these relations is based on computer algebra algorithms which involve the manipulation of univariate and multivariate polynomials. Gröbner bases are a core ingredient in these computations: they provide canonical generating sets with strong properties, upon which one can elaborate algorithmic solutions to many questions related to such algebraic relations.