I study differential manifolds and PDEs in sub-Riemannian setting. This is a geometric setting which extends Riemannian manifolds, but is totally degenerate.
Studies of sub-Riemannian manifolds started only recently. Open problems are due to the presence of so called characteristic points, where the normal is not defined. As a consequence notions like curvature are not well defined.
I also work on regularit of quasilinear subelliptic PDE, and in particular curvature PDE and geometric flows. The problem is more complex when regularity at the boundary is considered, due to the presence of characteristic points.
I also study mathematical models of the visual cortex. The lower areas are very geometric and can be described as Lie groups with a sub-Riemannian metric.
We will introduce a deep-learning structure directly inspired by the structure of the brain, and study existence of minima and stability
