# Research

## Gaining a mechanistic understanding of spatial biodiversity patterns

Biodiversity results from differentiation mechanisms developing within biological populations. Such mechanisms are influenced by the properties of the landscape over which individuals interact, disperse and evolve. The documentation of high levels of species diversity in complex mountain region or riverine systems suggest that some peculiar landscape properties foster diversity. I use mathematical models to investigate how connectivity and habitat heterogeneity foster diversity. My modeling approach complements empirical work on biodiversity by connecting documented patterns to the processes that have generated it. Preprint available here

## Developping numerical schemes for solving highly dimensional non-local nonlinear PDEs

Partial Differential Equations (PDEs) are equations that arise in a variety of models in physics, engineering, finance and biology. I develop numerical schemes based on machine learning techniques to solve for a special class of PDEs (cf below) in high dimension.

\begin{equation} \begin{aligned} (\tfrac{\partial}{\partial t}u)(t,x) &= \int_{D} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) \, \nu_x(d{\bf x}) \\ & \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{ 1 }{ 2 } \text{Trace}\!\big( \sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)( t,x ) \big). \end{aligned} \end{equation}

Such PDEs permit to e.g. model the evolution of biological populations in a realistic manner, by describing the dynamics of several traits characterising individuals (for a population of birds, think of traits as the beak length, the body size, or the color of the birds). I have implemented those schemes in HighDimPDE.jl, a Julia package that should allow scientists to develop models that better capture the complexity of life. These techniques extend beyond biology and are also relevant for other fields, such as finance. 