We will develop research in these three direction of research:

• Sub-Riemannian differential geometry in Lie groups.
  We first need to correctly define the properties of the objects of the Lie groups, which are regular curves, surfaces, manifolds and k-forms, with instruments of differential geometry.
• PDEs in Lie groups
  Once the basic objects of the space are understood, we will describe their motion in the geometrical structure, via partial differential equations. These equations are called sub-elliptic and ultra-parabolic, and are not elliptic or parabolic at any point. The graphs of the solutions, its level lines, its jump set, are regular objects in the previously defined sense.
• Models of the visual cortex in Lie groups
  The visual cortex will be modelled using Lie groups insturment: families of cells sensitive to different cognitive tasks will be studied, and each of them modelled with a different Lie group. Within each family we will describe the action of the receptive profiles, their long range connectivity and their spatial organization. We will also exploit interactions, feed forward and feed back process from one family of cells to the other, in order to have an insight of the mechanism of the whole cortex