PhD students working group

The Universe exhibits a web-like pattern known as the large-scale structure (LSS), where galaxies are assembled into clusters, filaments, and walls surrounding vast underdense regions called cosmic voids. This structure is not directly apparent in the night sky, nor in early astronomical observations, but emerged progressively as galaxy surveys increased in depth and size.
I will give a brief review of the historical development of our exploration of the LSS, starting from Hubble’s identification of galaxies beyond the Milky Way and the discovery of cosmic expansion through redshift measurements, and continuing to modern redshift surveys that map the three-dimensional distribution of millions of galaxies.
These advances revealed the cosmic web and voids as a dominant feature of the observed Universe. This leads to the cosmological survey data I will use for my PhD project and should illustrate the dramatic increase in scale and depth achieved by the next generation of surveys. Such advances require increasingly robust modelling and statistical tools to map and characterize the cosmic web and its voids. This provides the context for my work and, I hope, perhaps answers a question some of you may have been wondering over the past year: what exactly is a cosmologist doing at the IECL?

L’analyse topologique des données (Topological Data Analysis : TDA) est un domaine à l’intersection de la statistique et de la topologie algébrique qui a émergé au début des années 2000.

L’objectif est de tirer de nouvelles informations dans les données, en s’intéressant à des aspects géométriques et topologiques dans leur structure. Le cadre usuel consiste à étudier la structure d’un nuage de points, c’est-à-dire un ensemble de points dans un espace métrique, et un des outils le plus utilisé pour cela est ce qu’on appelle l’homologie persistante.

Dans cet exposé nous commencerons par introduire de manière pédagogique l’homologie persistante, puis nous discuterons de ces applications possibles dans un problème de neurosciences.

Cet exposé tourne autour de trois problèmes distincts mais fortement reliés. Tout d’abord, l’étude de la limite hydrodynamique de systèmes de particule soumises à des dynamiques de branchement et de sélection, qui est la question centrale que je me suis posée pendant ma thèse.

Ensuite, les équations de réaction-diffusion faisant intervenir une frontière libre contrôlant la masse totale, connus depuis une vingtaine d’année pour être reliés aux systèmes de particule en interactions.

Et enfin, le problème inverse du premier temps de passage pour un processus markovien, que l’on peut interpréter comme une reformulation probabiliste des problèmes à frontière libre.

Mon but sera de vous présenter ces trois problèmes et de vous expliquer l’état de la littérature sur ce qui les relient.

A first simplification of the gene expression mechanism considers that a gene is transcribed into messenger RNA, which in turn is translated into protein. Single-cell data have revealed the presence of biological variability between cells of identical genome and environment, highlighting not only epigenetic aspects but also the stochastic nature of gene expression.

In the context of regulatory networks underlying cell states and types, we need to build a model that takes into account both stochasticity and the interaction of genes with each other. Here we focus on a dynamical model of gene expression, formulated as a piecewise-deterministic Markov process (PDMP) and describing an arbitrary number of interacting genes. This stochastic model is able to reproduce the biological variability measured experimentally, but remains mathematically complex and difficult to study. This is why, in the litterature, a simplified model with only proteins is considered. 

During this talk, we provide insights on construction and use of semigroups and infinitesimal generators for PDMPs. Afterwards we present both models and use coupling methods to explicitly upper bound the error made when substituting the full model with its simplified version.

A fundamental question in geometry consists in understanding the effect of curvature on the shape of geometric spaces. In the case of surfaces, the Gauss-Bonnet Theorem establishes a link between the curvature of a surface and its topology. For example, it allows us to understand the topology of surfaces whose curvature is positive at every point.

When considering higher-dimensional geometric objects, called manifolds, we can define different notions of curvature. Scalar curvature is the weakest of these notions, and for this reason it is difficult to extract topological or geometric information from it. In particular, can we describe the topology of a manifold with positive scalar curvature?

In this talk, I will explain why this is an interesting question, and I will present a classification result for 3-dimensional manifolds with positive scalar curvature. This is a collaborative work with F. Balacheff and S. Sabourau.

In this talk, we begin by examining the application of curvature flows to a broad range of geometric problems. Following this, we introduce the essential geometric concepts required to understand these flows. Thereafter we focus on the mean curvature flow and its singularities. In particular, we give an intuitive and accessible proof of why singularities must occur if the initial surface is compact. After conducting a graphical analysis of various types of singularities, we describe how these singularities can be modeled by self-similar solutions of the mean curvature flow. Motivated by this, we conclude the presentation by exploring a current area of research: investigating the behavior of solutions that are in the proximity of such self-similar solutions.

Cela fait déjà plus de 150 ans que la recherche mathématique se casse les dents sur ce fameux problème appelé “Hypothèse de Riemann”. Portant sur les zéros non triviaux de la fonction Zêta de Riemann, elle est étroitement liée à la répartition des nombres premiers.