
Numerical optimal transport / inverse problems in optics. I am working on the numerical aspect of optimal transport, in particular in the semidiscrete setting (i.e. when transporting an absolutely continuous measure to a discrete one). I am also investigating applications to inverse problems arising in nonimaging optics. Figure: Simulation with a raytracer of the light projected on a wall after reflexion of an isotropic punctual source light on an optical compound (reflector surface) that we built.  
Isometric embeddings / Convex Integration theory. I am working with the Hevea Project on the realization of Nash's isometric embeddings based on the implementation of the Convex Integration Theory developped by Gromov in the 70s. I am particularly interested in the simplification of this Convex Integration Theory in order to make it more effective and provide a numerical tool (more) usable in practice to solve some nonlinear partial differential equations. Click here for a concrete application. Figure: inside of an isometric embedding of a flat torus (in the three dimensional space) that reveals a smoothfractal structure  
Geometric inference. The aim of geometric inference is to get robust estimations of the topological and geometric properties of a geometric object from an approximation, such as a finite point set. I have in particular been working on the stability of the (Federer) curvature measures, the regularity of the distance functions to compact sets, Voronoi Covariance Measures using distances to measures and have also investigated applications in digital geometry. Figure: Example of a finite point set from which one wants to infer geometric properties of the underlying object. 