My main research interest is Geometric Measure Theory. Currently I benefit from the European Union’s research and innovation programme Horizon 2020 through the Marie Skłodowska Curie Action n.752018, CuMiN (Currents and Minimal Networks). The introduction to CuMiN’s research project is below. For a simpler introduction to branched optimal transport, see MaddMaths (Italian).
The core of this project is Geometric Measure Theory and, in particular, currents and their interplay with the Calculus of Variations and Partial Differential Equations. Currents have been introduced as an effective and elegant generalization of surfaces, allowing the modeling of objects with singularities which fail to be represented by smooth submanifolds.
Classically, currents’ main application is the Plateau problem of area minimization under a boundary constraint.
In the first part of this project we propose new and innovative applications of currents with coefficient in a group to other problems of cost-minimizing networks typically arising in the Calculus of Variations and in Partial Differential Equations: with a suitable choice of the group of coefficients one can study optimal transport problems such as the Steiner tree problem, the irrigation problem (as a particular case of the Gilbert-Steiner problem), the singular structure of solutions to certain PDEs, variational problems for maps with values in a manifold, and also physically relevant problems such as crystals dislocations and liquid crystals. Since currents can be approximated by polyhedral chains, a major advantage of our approach to these problems is the numerical implementability of the involved methods.
In the second part of the project we address a challenging and ambitious problem of a more classical flavor, namely, the boundary regularity for area-minimizing currents. Our research program, which is modeled on the approach to the regularity of area-minimizing currents developed in the celebrated Almgren’s Big Regularity Paper and in the more recent papers by De Lellis and Spadaro, requires some of the most sophisticated analytical tools presently available.
In the last part of the project, we investigate fine geometric properties of normal and integral (not necessarily area-minimizing) currents. These properties allow for applications concerning celebrated results such as the Rademacher theorem on the differentiability of Lipschitz functions and a Frobenius theorem for currents.