Geometric evolution of curves, surfaces and networks

Optimization and evolution problems involving systems of curves and surfaces, and more generally networks or branched structures arise in a variety of contexts, embracing classical geometric problems such as mean curvature flow, Plateau problem and Steiner problem, as well as more general geometric evolution problems of physical interest (e.g. relativistic strings and membranes in cosmology, interface dynamics of grain boundaires and of immiscible multiphase fluids), and optimal (branched) transportation models in economics, biology, image processing and data analysis. Such problems attracted a lot of attention in recent years, and several refined tools from geometric analysis, partial differential equations and geometric measure thery were developed to handle, theoretically and numerically, suitable weak notions of solutions, and to analyse the presence or the formation of singularities within the models under investigation. In this project we will focus on three main topics, aiming at substantially improve the state-of-the-art.

1) Optimal transport and network optimization
We  provide new robust formulations of optimal transportation problems like Steiner tree or irrigation-type problems in any dimensions, using tools from geometric measure theory (e.g. currents), studying convex relaxations and giving efficient numerical schemes in any dimensions. We will also give robust variational approximations (gamma convergence) trough Ginzburg-Landau energies

2) Mean curvature flow of networks 
We study weak notions of global solutions for this essentially singular flow and selection principles in case of nonuniqueness after the formation of singularities. Besides Brakke varifold notion we will use the robust De Giorgi minimizing movements approach, so as to include forcing terms, inhomogeneities and anisotropies. 
We will also analyse Willmore-type elastic energies on networks, starting by proving existence of static minimizing configuration. 

3) Geometric evolutions of hyperbolic type 
We study minimal surfaces with respect to a Lorentzian metric (e.g. in Minkowski space-time) as models for relativistic strings and membranes in cosmology and gravitation, that account for quantum effects and topological defects in relativistic field theories.
Their time slices evolve according to a relativistic hyperbolic curvature flow that develops singularities in finite time (extinction, cusps). We aim to explore set theoretic solutions based on level set approach, which is robust with respect to possible topological changes occurring when singularities appear in the evolution, and also  also variational approaches like implicit time discretization schemes (Morse semi-flow) corresponding to the minimization of a suitable convex functional, inspired to the classical BMO scheme for the mean curvature flow.