CuMiN · Currents and Minimizing Networks

Currents and Minimizing Networks

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 one-dimensional currents with coefficients 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, 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 such as a Frobenius theorem for currents linking the rectifiability of a current and its boundary to an algebraic property of its tangent field, namely the involutivity. The project aims at having a significant social impact at several levels: the latter parts of the project (boundary regularity for area-minimizing currents and geometric properties of currents) are abstract and technical, but very important for the mathematical community, also from a historical point of view. The part of the project specifically concerning minimal networks has not only a scientific interest: its numerical versatility seems promising for the development of new algorithms for the computation of optimal networks (Steiner problem, Gilbert-Steiner problem and multimaterial transport problem). Moreover, the theme particularly fits interdisciplinarity (between Mathematics, Computer Science, Logistics and Urban Planning, Engineering) and popularization.

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