Publications

• Well, Better and In-Between

  with Raphaël Carroy. (2020) In Well Quasi-orders in Computation, Logic, Language and Reasoning. Ed. Peter Schuster, Monika Seisenberger, Andreas Weiermann, Trends in Logic, Springer.
  See abstract.
  Hide abstract. Starting from well-quasi-orders (wqos), we motivate step by step the introduction of the complicated notion of better-quasi-order (bqo). We then discuss the equivalence between the two main approaches to defining bqo and state several essential results of bqo theory. After recalling the role played by the ideals of a wqo in its bqoness, we give a new presentation of known examples of wqos which fail to be bqo. We also provide new forbidden pattern conditions ensuring that a quasi-order is a better quasi-order.

• Embeddability on functions: order and chaos

  with Raphaël Carroy and Zoltán Vidnánszky. Trans. Amer. Math. Soc. 371 (2019), 6711-6738. Preprint available on arXiv.org
  See abstract.
  Hide abstract. We study the quasi-order of topological embeddability on definable functions between Polish zero-dimensional spaces. We first study the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order.
  We then turn to the existence of maximal elements with respect to embeddability in a given Baire class. It is proved that the class of continuous functions is the only Baire class to admit a maximal element. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

• Towards better: a motivated introduction to BQO

  EMS Surveys in Mathematical Sciences, Volume 4, Issue 2, 2017, pp. 185-218.Preprint available on arXiv.org
  See abstract.
  Hide abstract. The well-quasi-orders (WQO) play an important role in various fields such as Computer Science, Logic or Graph Theory. Since the class of WQOs lacks closure under some important operations, the proof that a certain quasi-order is WQO consists often of proving it enjoys a stronger and more complicated property, namely that of being a better-quasi-order (BQO).
  Several articles contains valuable introductory material to the theory of BQOs. However, a textbook entitled "Introduction to better-quasi-order theory" is yet to be written. Here is an attempt to give a motivated and self-contained introduction to the deep concept defined by Nash-Williams that we would expect to find in such a textbook.

• Finite versus infinite: an insufficient shift

  Advances in Mathematics, 2017, 320 (7), 244-249. Preprint available on arXiv.org
  See abstract.
  Hide abstract. The shift graph is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form Gf defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f:X→X. This answers the primary outstanding question from (Kechris, Solecki and Todorcevic; 1999).

• A Wadge hierarchy for second countable spaces

  Archive for Mathematical Logic, 2015, 54 (5), 659–683. The final publication is available at Springer.
  See abstract.
  Hide abstract. We define a notion of reducibility for subsets of a second countable T0 topological space based on, relatively continuous relations and admissible representations. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T0 space, it yields a hierarchy on Borel sets, namely it is wellfounded and antichains are of length at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line or the Scott Domain P(ω).

• From Well to Better, the Space of Ideals

  Fundamenta Mathematicae, 2014, 227 (3), 247-270. With Raphaël Carroy. DOI: 10.4064/fm227-3-2
  See abstract.
  Hide abstract. On the one hand, the ideals of a well quasi-order (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, Nash-Williams' barriers are given a uniform structure by embedding them into the Cantor space.
  We prove that every map from a barrier into a wqo restricts on a barrier to a uniformly continuous map, and therefore extends to a continuous map from a countable closed subset of the Cantor space into the space of ideals of the wqo. We then prove that, by shrinking further, any such continuous map admits a canonical form with regard to the points whose image is not isolated.
  As a consequence, we obtain a simple proof of a result on better quasi-orders (bqo); namely, a wqo whose set of non principal ideals is bqo is actually bqo.

• Duality and the equational theory of regular languages

  Future Directions for Logic: Proceedings of PhDs in Logic III. IfColog Proceedings 2. College Publications.
  See abstract.
  Hide abstract. On one hand, the Eilenberg variety theorem establishes a bijective correspondence between varieties of formal languages and varieties of nite monoids. On the other hand, the Reiterman theorem states that varieties of finite monoids are exactly the classes of finite monoids definable by profnite equations. Together these two theorems give a structural insight in the algebraic theory of finite automata. We explain how duality theory can account for the combination of this two theorems, as it was pointed out by (Gehrke, Grigorieff and Pin, 2008).

Ph.D. Thesis

Better-quasi-order: Ideals and Spaces

Selected undergraduate works

• Détermination projective

  Master thesis (in french), EPFL, July 2010. Pdf

• Automates, Langages et Logique

  Semester Project (in french), EPFL, Autumn 2009. Pdf

• Théorie spectrale et évolution en mécanique quantique

  Semester Project (in french), EPFL, Autumn 2008. Pdf