## Periodic Reporting for period 1 - CuMiN (Currents and Minimizing Networks)

Reporting period: 2017-09-01 to 2019-08-31

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.

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.

"Accepted/published papers:

(1) Marchese-Massaccesi-Tione: A multi-material transport problem and its convex relaxation via rectifiable G-currents.

(2) Don-Massaccesi-Vittone: Rank-one theorem and subgraphs of BV functions in Carnot groups.

(3) Colombo-De Lellis-Massaccesi: The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system.

(4) De Lellis-De Philippis-Hirsch-Massaccesi: Boundary regularity of mass-minimizing integral currents and a question of Almgren. In this preoceedings we give an overview of the result in (5) and its consequences.

Submitted papers:

(5) De Lellis-De Philippis-Hirsch-Massaccesi: On the boundary behavior of mass-minimizing integral currents.

(6) Alberti-Massaccesi-Stepanov: On the geometric structure of normal and integral currents, II.

(7) Marchese-Massaccesi-Stuvard-Tione: A multi-material transport problem with arbitrary marginals.

(8) Carioni-Marchese-Massaccesi-Pluda-Tione: The oriented mailing problem and its convex relaxation.

Seminars:

• Seminar PDE and Mathematical Physics, Zurich.

• Mathematical Analysis Seminar, Rome Tor Vergata.

• Mathematical Analysis Seminar, Padova.

• Rough differential calculus and weak geometric structures, Moscow.

• Joint seminar Department of Mathematics – Centro Internazionale per la Ricerca Matematica, Trento.

• Meeting in Applied Mathematics and Calculus of Variations, Roma.

• Metric Analysis and Regularity, Catania.

• Winter school on FLUid DYnamics, DIspersive equations and QUAntum fluids, Bressanone.

• 10th Itinerant Workshop in PDE’s, Roma.

• MathBites, Trento.

• Workshop on Calculus of Variations, Brighton.

• Workshop on Geometric Analysis and Geometric Measure Theory, Pisa.

• Journey of Women in Math, L’Aquila.

• Brescia-Trento Nonlinear Day, Trento.

• GMT and PDEs in Basel.

• Mathematical Analysis Seminar, Aachen.

Events:

• Organization of the workshop “Geometric Measure Theory in Verona”, June 10-15, 2018.

• Research in Pairs (CIRM), Trento (with G. Alberti, A. Marchese e D. Vittone).

• Organization of the Intensive Meeting “Geometric Measure Theory” in Canazei, June 26-29, 2019.

Supervision of Cuong's master thesis on ""Branched Optimal Transport in a Riemannian manifold""

Ph.D. course on ""Regularity of Elliptic Partial Differential Equations"" at the University of Verona.

Popularization:

• Publication of the article “Il trasporto ramificato: uno... sconto comitiva” in the mathematical popularization website http://maddmaths.simai.eu/

• Laboratories ""La matematica delle bolle di sapone"" in San Giovanni Lupatoto and KidsUniversity's initiatives in Verona and Padova."

(1) Marchese-Massaccesi-Tione: A multi-material transport problem and its convex relaxation via rectifiable G-currents.

(2) Don-Massaccesi-Vittone: Rank-one theorem and subgraphs of BV functions in Carnot groups.

(3) Colombo-De Lellis-Massaccesi: The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system.

(4) De Lellis-De Philippis-Hirsch-Massaccesi: Boundary regularity of mass-minimizing integral currents and a question of Almgren. In this preoceedings we give an overview of the result in (5) and its consequences.

Submitted papers:

(5) De Lellis-De Philippis-Hirsch-Massaccesi: On the boundary behavior of mass-minimizing integral currents.

(6) Alberti-Massaccesi-Stepanov: On the geometric structure of normal and integral currents, II.

(7) Marchese-Massaccesi-Stuvard-Tione: A multi-material transport problem with arbitrary marginals.

(8) Carioni-Marchese-Massaccesi-Pluda-Tione: The oriented mailing problem and its convex relaxation.

Seminars:

• Seminar PDE and Mathematical Physics, Zurich.

• Mathematical Analysis Seminar, Rome Tor Vergata.

• Mathematical Analysis Seminar, Padova.

• Rough differential calculus and weak geometric structures, Moscow.

• Joint seminar Department of Mathematics – Centro Internazionale per la Ricerca Matematica, Trento.

• Meeting in Applied Mathematics and Calculus of Variations, Roma.

• Metric Analysis and Regularity, Catania.

• Winter school on FLUid DYnamics, DIspersive equations and QUAntum fluids, Bressanone.

• 10th Itinerant Workshop in PDE’s, Roma.

• MathBites, Trento.

• Workshop on Calculus of Variations, Brighton.

• Workshop on Geometric Analysis and Geometric Measure Theory, Pisa.

• Journey of Women in Math, L’Aquila.

• Brescia-Trento Nonlinear Day, Trento.

• GMT and PDEs in Basel.

• Mathematical Analysis Seminar, Aachen.

Events:

• Organization of the workshop “Geometric Measure Theory in Verona”, June 10-15, 2018.

• Research in Pairs (CIRM), Trento (with G. Alberti, A. Marchese e D. Vittone).

• Organization of the Intensive Meeting “Geometric Measure Theory” in Canazei, June 26-29, 2019.

Supervision of Cuong's master thesis on ""Branched Optimal Transport in a Riemannian manifold""

Ph.D. course on ""Regularity of Elliptic Partial Differential Equations"" at the University of Verona.

Popularization:

• Publication of the article “Il trasporto ramificato: uno... sconto comitiva” in the mathematical popularization website http://maddmaths.simai.eu/

• Laboratories ""La matematica delle bolle di sapone"" in San Giovanni Lupatoto and KidsUniversity's initiatives in Verona and Padova."

In (1) we cast a new transport problem which takes into account the virtuous interaction between different materials that can be transported together. Behind the complete novelty of the problem, we prove the existence of the solution and the equivalence with a mass-minimization problem for currents with coefficients in a suitable group. In (7), we are able to analyze the continuous case and the relaxation of the cost to normal currents. This work has a potentially high impact on the engineering community, because some (already available) technologies build on the exploitation of a network for different purposes (see, for instance, the powerline communications technology and the oriented mailing problem in (8)).

The submission of the paper “On the boundary behavior of mass-minimizing integral currents” (5) marks the completion of the second part of the project, establishing a regularity result that arises great interest and curiosity in the scientific community. As a byproduct of the efforts for this regularity result, we gained a knowledge of the regularity techniques (for area-minimizing currents and elliptic PDEs) which allowed us to successfully attack the regularity problem for solutions to the hyperdissipative Navier-Stokes equations and establish the best estimate for the Hausdorff dimension of the singular set currently available (see (3)).

Concerning the last part of the project, in (6) we present a complete picture of the behavior of normal currents with respect to the geometric property of the boundary. Further investigations on the generalisations of Frobenius theorem are the subject of an ongoing collaboration with G. Alberti and A. Merlo.

In the frame of the attempts to investigate fine properties of normal currents and the sharpness of Rademacher theorem, in 2016 D. Vittone and I found a very short and elegant proof of the celebrated rank-one theorem for BV functions by G. Alberti (BV functions are top-dimensional normal currents). In (2) we extend the (previously only conjectured) theorem to Carnot groups with certain algebraic properties.

The submission of the paper “On the boundary behavior of mass-minimizing integral currents” (5) marks the completion of the second part of the project, establishing a regularity result that arises great interest and curiosity in the scientific community. As a byproduct of the efforts for this regularity result, we gained a knowledge of the regularity techniques (for area-minimizing currents and elliptic PDEs) which allowed us to successfully attack the regularity problem for solutions to the hyperdissipative Navier-Stokes equations and establish the best estimate for the Hausdorff dimension of the singular set currently available (see (3)).

Concerning the last part of the project, in (6) we present a complete picture of the behavior of normal currents with respect to the geometric property of the boundary. Further investigations on the generalisations of Frobenius theorem are the subject of an ongoing collaboration with G. Alberti and A. Merlo.

In the frame of the attempts to investigate fine properties of normal currents and the sharpness of Rademacher theorem, in 2016 D. Vittone and I found a very short and elegant proof of the celebrated rank-one theorem for BV functions by G. Alberti (BV functions are top-dimensional normal currents). In (2) we extend the (previously only conjectured) theorem to Carnot groups with certain algebraic properties.