PhD ThesisHere is my PhD thesis, Functions of the first Baire class (pdf), jointly supervised by Jacques Duparc from the University of Lausanne and Olivier Finkel from the University Paris 7 - Denis Diderot. It gathers the articles Playing in the first Baire class and A quasi-order on continuous functions. The general summary at the beginning is in french, the remainder of the thesis is in english.
Published or accepted papers
Sigma-continuity with closed witnesses (joint work with Benjamin D. Miller)
Fundamenta Mathematicae, 237, pp. 29-42, 2017 (pdf)
We use variants of the G0 dichotomy to establish a refinement of Solecki's basis theorem for the family of Baire-class one functions which are not σ-continuous with closed witnesses.
Epimorphisms between linear orders (joint work with Riccardo Camerlo and Alberto Marcone)
Order, 32, pp.387-400, 2014 (pdf)
We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.
To respect the copyright agreement signed with Order, the file available on this page is a preprint. The published version of this article is available on Order's website or on demand by mail.
From well to better, the space of ideal (joint work with Yann Pequignot).
Fundamenta Mathematicae, 227, pp 247-270, 2014 (pdf)
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.
Playing in the first Baire class
Mathematical Logic Quarterly, volume 60, n. 1-2, pp. 118-132, 2014 (DOI 10.1002/malq.201200064) (pdf).
We present a self-contained analysis of some reduction games, which characterise various natural subclasses of the first Baire class of functions ranging from and into 0-dimensional Polish spaces. We prove that these games are determined, without using Martin's Borel determinacy, and give precise descriptions of the winning strategies for Player I. As an application of this analysis, we get a new proof of the Baire's lemma on pointwise convergence.
To respect the copyright agreement signed with MLQ, the file available on this page is a preprint. The published version of this article is available on MLQ's website or on demand by mail.
A quasi-order on continuous functions
Journal of Symbolic Logic, volume 78, n. 2, pp. 633-648, 2013. (pdf)
We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.
Decomposing Baire class one functions
Future Directions for Logic, Proceedings of PhD's in Logic III, pp 1--11, 2012.
This is a conference paper that prepares Playing in the first Baire class.
Articles either submitted or in preparation
Topological embeddability between functions: order and chaos (joint work with Yann Pequignot and Zoltán Vidnyánszky)
Submitted, 2018 (pdf)
Well, Better and In-Between (joint work with Yann Pequignot)
Submitted, 2017 (pdf)
Linear orders: when embeddability and epimorphisms agree (Joint work with Riccardo Camerlo and Alberto Marcone)
The open dihypergraph dichotomy and the second level of the Borel hierarchy (Joint work with Benjamin D. Miller and Dániel T. Soukup)
Descriptive Cosmology (Joint work with Johan Comparat)
Bases for functions beyond the first Baire class (Joint work with Benjamin D. Miller)
Some advice on a first article, gathered here and there
There is no real reason to let someone in the dark for his/her first article (in english), when some tricks are well-known! I have gathered some here, please feel free to comment/correct/add.
Making a tree with a function?
Following a talk, Taras Banakh suggested a quasi-order between functions that is bqo for simple reasons. I explain here on an example why it does not respect the Borel complexity of preimages, or Borel degree function. (pdf)