In collaboration with Miraine Davila and Josué Corujo (both based in France), we study the evolution of connected components in random graphs that grow through the sequential addition of vertices. Surprisingly, the resulting dynamics reveal deep connections between two seemingly unrelated areas: the Fleming–Viot process from population genetics and the multiplicative coalescent from random graph theory. This project aims to better understand how genealogical and coalescent structures emerge in growing networks.

Inspired by the coloring-rule techniques introduced in recent work [12], we are developing a multidisciplinary framework that combines probability theory, ecology, evolutionary biology, and data science to infer competition networks from complex ecological data. By uncovering the interactions that structure biological communities, we aim to develop quantitative tools that can support conservation efforts, identify keystone species, and improve our understanding of the mechanisms that sustain biodiversity and ecosystem stability.

The population genetics of plasmids and endosymbionts presents a rich source of mathematical challenges arising from the interaction of evolutionary forces across multiple biological scales. Our research combines stochastic processes, population genetics, microbiology, and experimental evolution to understand how genetic variation is generated, maintained, and transmitted in these systems. By developing mathematical models informed by experimental data, we seek to uncover the mechanisms governing adaptation, persistence, and evolution in plasmid-bearing bacteria and host–symbiont associations. Particular emphasis is placed on the interplay between within-host and between-host dynamics, and on understanding how these interactions shape the long-term evolutionary fate of mobile genetic elements and symbiotic organisms. Representative contributions include [35, 22].

Moment duality has been a central theme of my research [34,28, 27,19,14,11,7]. In recent work with Ariel Offenstadt and Arno Siri-Jégousse, we have explored the deep connections between exchangeability and duality, using ideas from de Finetti theory to gain a better understanding of dual stochastic processes. This perspective has led to new constructions and characterizations of dualities arising in population genetics and interacting particle systems.

One of our long-term goals is to build a mathematical theory of experimental evolution. We develop probabilistic and population genetic models that connect evolutionary mechanisms, such as mutation, selection, and clonal interference, with observations from laboratory evolution experiments. This research program includes a mathematical analysis of the Long-Term Evolution Experiment, recognized with the Ito Prize in 2017 [6], and a data-driven study of evolutionary trajectories, awarded the Feldman Prize in 2022 [8]. And more recently [32], which deals with clonal interference.

Dormancy has been a long-standing research focus of our group. In [5], we introduced the Seed Bank Coalescent, a genealogical framework that captures the effects of dormancy on genetic diversity. Since then, we have continued to develop the theory and its applications, studying topics ranging from population genetics and evolution to stochastic processes and coalescent theory.

Interacting particle systems are among my favorite research topics. Viewing these models through the lens of population genetics and stochastic duality has often provided new insights and useful techniques. Our work includes contributions to the contact process [ 31], recent advances on the symmetric exclusion process, and ongoing projects on the voter model. More broadly, we are interested in understanding how local interactions give rise to large-scale phenomena in complex stochastic systems.

This research project forms the core of Imanol Nuñez's PhD thesis. Under the co-supervision of José Luis Pérez (CIMAT) and in close collaboration with Noemi Kurt (Frankfurt), we investigate the relationship between exchangeability and moment duality. By studying exchangeable Markov chains and their associated genealogical structures, we aim to develop a deeper understanding of duality and its role in probability theory and population genetics.

This is my main collaborative project with Johnny Yang, although he is involved in many other exciting research directions. Together, we study the emergence of stochastic partial differential equations (SPDEs) from individual-based models, with a particular focus on universality phenomena. One of our goals is to understand the broad class of microscopic systems whose large-scale behavior is governed by equations such as the Fisher–Kolmogorov–Petrovsky–Piskunov (F-KPP) equation. This project is also a collaboration with Louis Fan (University of North Carolina at Chapel Hill).