CUMIN - Currents and Minimizing Networks

Starting date
September 1, 2017
Duration (months)
24
Departments
Computer Science
Managers or local contacts
Orlandi Giandomenico

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.
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. 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.
The Marie Skłodowska-Curie fellowship and the subsequent possibility of a close collaboration with Prof. Orlandi are
a great opportunity of fulfillment of my project, which is original and independent but is also capable of collecting the
best energies of several young collaborators.

Sponsors:

Comunità Europea
Funds: assigned and managed by the department

Project participants

Annalisa Massaccesi
Research Scholarship Holders
Research areas involved in the project
Matematica - applicazioni e modelli
Calculus of variations and optimal control; optimization - -

Activities

Research facilities