My PhD is codirected by Frédéric Chapoton and Christian Stump. It started in September, 2020
The main project is about two generalizations of the Tamari lattice, namely the $m$-Tamari and $m$-Cambrian lattices. In both cases, $m$ is a positive integer and in the case $m = 1$ we recover precisely the Tamari lattice. The starting point is a conjecture of Christian Stump, Hugh Thomas and Nathan Williams that states that the $m$-Tamari lattice and the type A linear $m$-Cambrian lattice would have the same number of intervals. Moreover, in a work of Mireille Bousquet-Mélou, Eric Fusy and Louis-François Préville Ratelle, they count the number of intervals of the $m$-Tamari lattice.
My goal is now to count the intervals in the $m$-Cambrian lattice, maybe by finding a bijection between both intervals or by proving that their generating series satisfy the same equations. Here is a poster about this topic.
I started this project in April, 2022. It starts from an observation of Frédéric that two lattices defined on Dyck paths seemed to have the same number of linear (totally ordered) intervals, even when distinguished according to their height.
During this work, I described the structure of the linear intervals, which led to a combinatorial decomposition of linear intervals and then I solved the equation to get the number of intervals. The next goal is to generalize this result to a bigger family of posets. Here is a poster that presents this project.
As a Master student I went to the UBO (Université de Bretagne Occidentale), in Brest, for an internship under the direction of Johannes Huisman. This work had to do with algebraic geometry, and more specifically about realizations of pseudo-lines configurations.
Given a real algebraic curve, one can define its pseudo-lines and pseudo-ovals. When the union of its pseudo-lines constitute an arrangement, we call it an algebraic pseudo line arrangement. The combinatorics of their intersection is encoded in a lattice. Given such a lattice, it may be realizable but by curves with a very high degree. In this work, we were interested by curves of low degree and we defined the class of curves with many pseudo lines. The question raises wether the lattice of a given line arrangement is algebraically realisable by a curve with many pseudo lines or not.
We proved that it was true for an infinite family of nontrivial lattices.
As a Bachelor student, I was at the CMLA in the ENS Cachan, for an internship under the direction of Enric Meinhardt-Llopis and Marie d'Autume. The goal was to extrapolate the values of a function outside of a domain (continuous or discrete), especially for reconstructing images, which can be useful for image compression, for instance.
The approach was to solve a biharmonic equation, then to relax the border condition with a penalty weight so that the solution could slightly differ from the data points, in order to improve the results. Different penalties and variations of the algorithm were studied during this work.