Temporary teaching and research agent in pure mathematics

Curriculum vitae

I’m working in arithmetic geometry. More precisely, I study overconvergent de Rham–Witt cohomology and its coefficients. I’ve also become interested in applications of these theories in point-counting algorithms.

I have defended my PhD thesis in 2020.

Articles

1. Pseudovaluations on the de Rham–Witt complex, Bulletin de la Société Mathématique de France 150 (2022), no. 1, pp. 53–75, doi:10.24033/bsmf.2844

   For a polynomial ring over a commutative ring of positive characteristic, we define on the associated de Rham–Witt complex a set of functions, and show that they are pseudovaluations in the sense of Davis, Langer and Zink. To achieve this, we explicitly compute products of basic elements on the complex. We also prove that the overconvergent de Rham–Witt complex can be recovered using these pseudovaluations.

Preprints

1. Local structure of the overconvergent de Rham–Witt complex (2023), 54 pages, arXiv:2311.15449

   We give a general description of the structure of the relative de Rham–Witt complex on a polynomial ring, seen as an algebra over its integral part. After giving a control of the overconvergence of Lazard’s morphism, we similarly give the structure of the overconvergent complex for a finite étale algebra over a perfect Noetherian ring. We then deduce a generalization of the usual decomposition as a direct sum, which is compatible with overconvergence on the projections.

2. Faster addition of Witt vectors over a polynomial ring (draft), SageMath code

   We describe an algorithm which adds two Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithms uses an isomorphism of Illusie in order to do the addition in an adequate free module. We also give an implementation of the algorithm in SageMath, which turns out to be faster that Finotti’s algorithm, which was until now the most efficient one for this operation.

3. Using de Rham–Witt cohomology in Kedlaya’s algorithm (draft), SageMath code

   We explain how to replace Monsky–Washnitzer cohomology with overconvergent de Rham–Witt cohomology in Kedlaya’s algorithm in the case of hyperelliptic curves over a finite field of odd characteristic. This method yields a simpler formula for the action of the Frobenius. We describe how to construct cohomological reduction formulae allowing us to compute the zeta function of the curve. Finally, we give an implementation in SageMath of the algorithm.

4. Isocrystals and de Rham–Witt connections (draft)

   This paper contains the proof of my PhD thesis’ main theorem, which describes the category of locally free isocrystals with a Frobenius structure as a category of integrable overconvegrent de Rham–Witt connections.

Lecture notes

1. p-adic cohomology theories and point couting, draft updated on June 26, 2023, comments welcome!
2. Introduction à la théorie des groupes, online version updated on April 03, 2024.

How to spell my name

I have noticed that my name is often misspelled. This is because of its two diacritical signs, but mostly because of the two dashes in my family name. It is indeed a double dash, and not a simple one, nor a long one, and certainly not a typo!

If you use LaTeX2e with a version released after 2018, you simply have to use that code:

Rubén Muñoz-\relax-Bertrand

In Plain TeX, or for obsolete versions of LaTeX2e, you can use the following code:

Rub\'en Mu\~noz-\relax-Bertrand

I would be very grateful if you use the proper spelling of my name.