This page was written in 2019 and is due for an update!
My research interests are centered on two themes:
Below is a list of more specific topics on which I had worked as of 2019.
Graph-valued Markov chains are a very natural and convenient way to model dynamic networks. As a result, they are relevant in a variety of applications. I have studied three specific models of graph-valued Markov chains:
For each of these models, a coalescent approach makes it possible to study the stationary graph and to get explicit expressions for graph invariants such as the degree distribution, the number of connected components, etc. In the case of [6], it is also possible to show that there is a cut-off for the mixing time, i.e. that the chain goes to its stationary distribution in a very narrow window of time.
I am interested in random trees and, more generally, in networks resulting from processes where lineages split and merge through time. These are conveniently represented by mathematical structures such as directed acyclic graphs (DAGs).
I am currently working with Amaury Lambert and Mike Steel on a phylogenetically relevant class of DAGs (see below).
I am working with Amaury Lambert on oriented first passage percolation on the Bernoulli site percolation cluster of the hypercube. We have not yet been able to prove what we want, but while working on this topic I became interested in more general questions concerning oriented percolation in randomly oriented graphs [5].
Tree-child networks are among the most studied classes of phylogenetic networks. Unfortunately, their precise enumeration is still an open question there is no known algorithm to sample them uniformly at random. As a result, very little is known about the structure of typical tree-child networks.
Endowing tree-child networks with a "ranking" structure yields a combinatorial class of phylogenetic networks that are both easy to count and easy to sample uniformly. I am working on understanding the structure of typical large ranked tree-child networks with Amaury Lambert and Mike Steel.
Defining the notion of species has been and remains a controversial issue in biology. There is some consensus in favour of a definition based on the capacity of individuals to interbreed, but there is no theoretical model to tell us what "interbreeding networks" should be look like. Amaury Lambert, Florence Débarre and I have suggested and studied one such model in [3].
My research on structured population has been centered on two topics: matrix population models and measures of generation time.
Matrix population models are demographic models describing the dynamics, in discrete time, of a population structured into discrete classes. The system is governed by an equation of the form n(t + 1) = A n(t), where n is the vector tracking the abundances of the classes and A is a primitive matrix known as the projection matrix. These models are widely used, both by conservation biologists for real-world applications and by theoretical ecologists when they need mathematical tractability.
To every projection matrix A is associated a matrix P corresponding to the transition matrix of the Markov chain tracking the classes visited by a gene when we look at its ancestry backwards in time. Stéphane Legendre and I have used this Markov chain to study the generation time and the elasticities of the growth rate to the entries of the projection matrix (see [1] and below). David McCandlish and I have used the fact that every projection matrix A can be uniquely decomposed into a triplet (P, w, λ) – where w is the stable distribution of the population and λ is the asymptotic growth rate – to give an optimal way to "lump" classes of a matrix population model together, i.e. to reduce its number of classes with a minimal impact on the statistics of the population [2]. I am also working on a method that uses this decomposition to assess the impact of the differences between the vital rates of the individuals within a class on the statistics of the population.
The generation time is one of the most basic and widely used statistics of populations, but it is more of an intuitive notion than a well-defined concept and there are several ways to quantify it. It is therefore important to understand the advantages and shortcomings of each measure.
One of the most accepted measures consists in averaging the age of mothers over all births that happen during a fixed period of time. Stéphane Legendre and I have showed that, in the framework of matrix population models, this also corresponds to the average time between two reproductive events in the ancestral lineage of a gene, and we have derived a very simple expression for this quantity [1]. Even though we were primarily interested in the generation time, our main result is that classic quantities known as the elasticities of the growth rate to the entries of the projection matrix can be interpreted as asymptotic frequencies of the transitions of the model in the lineage of a gene.
Another widely used measure of generation time is obtained by averaging the age of mothers over all offspring produced by a cohort. However, there are several ways to compute this average and this has been a source of confusion [4].
Many bipartite ecological networks, such as pollination networks, have a high nestedness (meaning that plants that are pollinated by highly specialized pollinator species also tend to be pollinated by more generalist pollinators). Because the nestedness computed on these networks is much higher than that computed on bipartite Erdős-Rényi random graphs, the origin of nestedness has been a puzzle in recent years and has generally been attributed to specific effects such as preferential attachment.
I am working with Amaury Lambert and Tristan Lazard on a null-model to explain the formation of ecological networks that would serve as a better reference than the bipartite Erdős-Rényi random graphs used by many biologists. Our model is minimalistic (it does not include any effect such as preferential attachment) but has a higher nestedness than bipartite Erdős-Rényi random graphs, this suggests that the high nestedness of real-world ecological networks might not be that surprising.
During my master's, I have worked on a model for the evolution of altruism in a spatial continuum with Erol Akçay. Unfortunately, this model proved to be untractable and we had to rely exclusively on simulations. Although I am not actively working on this topic anymore, I am still very interested in this biological question and the mathematical challenges it raises, and I am discussing with Raphaël Lachièze-Rey about the possibility of adapting results concerning boolean models to come up with a tractable model.