Nonlinear partial differential equations describing FROnt propagation, Geometric variational problems and Singularities.

This project will focus on some issues regarding front propagation in reaction-diffusion systems, geometrical problems with a variational structure and solutions of elliptic and parabolic problems exhibiting singularities.

These are important problems in the vast field of (nonlinear) Partial Differential Equations which are motivated by Physics and Biology. The first part of the project will be devoted to the study of reaction-diffusion systems.

An important type of solutions for these systems are fronts which, in many situations, play a distinctive role in the long-time dynamics.

However, these issues are not well understood for some important classes of reaction-diffusion systems due to the non-applicability of the more standard and widely used mathematical tools. Our purpose will be to fill this gap. The second part will be devoted to several problems in geometry.

These problems are formulated within the framework of Geometric Measure Theory (GMT) and can be understood through a PDE approximation of Allen-Cahn or Ginzburg-Landau type.

Following this approach, we will tackle several of these problems which remain unsolved and the outcome will be of interest to both the Geometry and the PDE communities.

The third part, also intimately related to GMT, will be centered on the classical topic of singularities of harmonic maps and Ginzburg-Landau equations. We will study the existence of new types of solutions with distinguished behavior.

The second and the third part of the project are intimately interconnected, and their interactions have been explored for a long time.

However, the project also aims at establishing new relations between the first and third part, whose interactions have not been studied in depth so far. The project is autonomous but it has been enriched by several collaborators. The role of Prof.

Orlandi as supervisor is particularly adapted to the project's purposes and it will be crucial for its correct and effective development.