The mean curvature flow (MCF) describes the evolution of a hypersurface in time, where the velocity at each point is given by its mean curvature vector (i.e., the unit normal vector multiplied by the mean curvature). When initiated with a sphere in Rn, the MCF will shrink it homothetically to a point in finite time. In this talk, we introduce an adaptation of the mean convex MCF within the Heisenberg group setting. Our initial objective was to explore potential connections between this flow and the Heisenberg isoperimetric problem. Wewill discuss the existence and uniqueness of solutions and prove that the Pansu sphere does not evolve homothetically under the MCF. This work is based on joint research with Gaia Bombardieri and Mattia Fogagnolo.[-]

Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, \mathbb{C})$ is obtained as the holonomy of a branched projective structure. We will show that one of the central properties of complex projective structures, namely the complex analytic structure of their moduli spaces, extends to the branched case.[-]