I think he is more applied than pure. But he is a PDE guy. Like PL Lions who also got a Fields medal, Both are from Paris VI which is a top university. I met PLL a few times at IRIA (now INRIA) when he was about 21 because his dad was the legendary JL Lions and founder of numerical analysis in France.He's a pure mathematician dressing like that, he's probably got Asperger's, give him a break.
BTW PLL was the inventor of the viscosity method that is used for nonlinear PDEs in finance, e.g. the UVM model.
Politics will be the ruin of him. It's like brilliant academics wanting to become Provost.
edit: on Mathematical Geneaology, I see PLL and CV had the same supervisor.