Research interest -- Most mathematical models introduced in the life and social sciences literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations. These models, however, leave out any spatial structure or stochastic component, two factors that have been identified as key factors in how communities are shaped. The aim of my research is to understand the role of space and stochasticity in a wide variety of applied sciences such as ecology, epidemiology, population genetics, opinion and cultural dynamics, through the mathematical analysis of stochastic processes known as interacting particle systems. In these models, members of the population (particles) such as atoms, cells, plants or agents, are located on the set of vertices of a connected graph. The latter has to be thought of as a network of interactions that dictates the dynamics of the system as particles interact only locally with their neighbors, thus modeling the presence of a spatial structure. The main objective of research in this area is to understand the macroscopic behavior and spatial patterns that emerge from the microscopic interactions that describe the local dynamics of large systems. Space in this context must be understood in a broad sense: an edge between two vertices of the underlying graph may be synonymous of geographic proximity, but also friendship relation, adherence to the same political party, etc.

Impact in mathematics and applied sciences -- The field of interacting particle systems is simultaneously one of the most challenging topics of probability theory and a popular modeling tool in applied sciences. From the point of view of mathematics, the tradition is to establish analytical results about simple existing models, while from the point of view of applied sciences, the emphasis is on the development of realistic models and predictions based on numerical simulations, whereas it is known from past research that spatial simulations are difficult to interpret and might lead to erroneous conclusions. My research is a combination of these two aspects. The challenge is to introduce models that are both mathematically tractable and able to capture the essence of biological and sociological systems through fundamentally new mathematical features, as opposed to straightforward generalizations of existing models including a large number of parameters that are artificially complex. Interestingly, while this approach is highly motivated by applied sciences, it also gives rise to problems which, regardless of their applications, are important mathematically, and to challenging proofs that usually consists of subtle combinations of measure theory, probability theory (percolation, random walks, large deviations), graph theory and combinatorics.

Towards dynamic graphs and hypergraphs -- The traditional framework of interacting particle systems focuses on stochastic dynamics on static graphs. My recent research extends this framework in two directions, again motivated by phenomena that are ubiquitous in nature, by considering evolutions either on dynamic graphs or on hypergraphs. Dynamic graph structures arise naturally when looking at systems of particles that have the ability to shape their spatial environment, which results in systems involving dynamics on the graph but also dynamics of the graph, with a feedback between dynamics on and of the graph. Hypergraph structures are employed to model systems including an intermediate mesoscopic scale, which results in systems in which all the vertices of the same hyperedge are simultaneously updated.