Mini-symposium

Length Optimization

in the frame of the International Conference “Mathematics Days in Sofia – 2026”

This mini-symposium is devoted to one-dimensional length-optimization problems at the intersection of combinatorics, the calculus of variations, and analytic geometry. The topics include, but are not limited to:

  1. The Steiner tree problem
  2. The Gilbert–Steiner problem
  3. Branched optimal transport
  4. Maximal and average distance minimizers
  5. Extremal currents and mass-minimizing currents
  6. Planar soap films (bubbles)

Organizing Committee

  • Danila Cherkashin, Institute of Mathematics and Informatics – BAS, Bulgaria
  • Yana Teplitskaya, Université Paris-Saclay, France

Participants

  • Alexey Gordeev, Alfréd Rényi Institute of Mathematics, Hungary

  • Andrea Marchese, University of Trento, Italy

  • Antoine Prouff, Purdue University, USA

  • Antonio de Rosa, Bocconi University, Milano, Italy

  • Danila Cherkashin, Institute of Mathematics and Informatics, Bulgaria

  • Emanuele Paolini, Università di Pisa, Italy

  • Gloria Paoli, Università degli Studi di Napoli Federico II, Italy

  • Kristiyan Vasilev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria

  • Matteo Novaga, Università di Pisa, Italy

  • Paul Pegon, Université Paris-Dauphine, France

  • Pavel Prozorov, Saint Petersburg State University, Russia

  • Srikanta Karthik, Rutgers University, USA

  • Yana Teplitskaya, Université Paris-Saclay, France

Program and Abstracts

link to abstract

In this seminar, I will present two new models in branched optimal transport. The first introduces anisotropy, assigning different costs to trajectories depending on their orientation. The second favors redundancy of paths, with the goal of increasing the robustness of the network against possible damage while preserving efficiency. Finally, I will discuss a recent extension to all regimes of the stability property for the mailing problem.

Based on joint works with Martina Bellettini, Luigi De Masi, Jakub Krukowski, and Annalisa Massaccesi.

In branched transportation, minimizers of the irrigation problem provide a cheapest way to connect two probability measures with a one-dimensional network of roads. We study the Lagrangian version of this problem, where the transport plan is described by trajectories, as opposed to the Eulerian version which gives only the pointwise velocities along the roads. We prove that the Lagrangian optimizers are stable, in the sense that the limit of a converging sequence of optimizers is still an optimizer. To do so, we establish that there is no loss of information between the limiting optimizers in the Lagrangian and Eulerian models, and therefore reduce the problem to an earlier known stability result in the Eulerian formulation. Joint work with Maria Colombo, Antonio De Rosa, Andrea Marchese and Paul Pegon.

We discuss recent progress on the structure and rigidity of volume-constrained critical points for anisotropic surface energies, capillary functionals, and the k-bubble problem.
For anisotropic and capillary energies, we work in the general class of finite perimeter sets and show that critical points exhibit strong rigidity, being characterized by unions of canonical minimizing shapes under minimal assumptions.
For the k-bubble problem, where multiple chambers interact, we require convexity of the chambers in order to analyze the geometry of the interfaces and obtain a classification of stationary configurations.

The Steiner Problem consists of connecting a given set of points with a graph of minimal length. We consider an extension of the problem that includes the possibility of having infinitely many given points. We will describe general existence and regularity results, as well as two examples of solutions to the extended problem with countably and uncountably many points.

This comprises joint works with Danila Cherkashin, Eugene Stepanov, and Yana Teplitskaya.

link to abstract

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We consider an isoperimetric problem for planar tilings with possibly unequal cells. We present general existence and regularity results, and study the classification of planar isoperimetric double tilings, namely tilings with two repeating cells of minimal perimeter. In this case, we explicitly determine the associated energy profile and provide a complete description of the phase transitions. We also comment on possible extensions and open problems.

This is based on joint work with F. Nobili and E. Paolini.

link to abstract

link to abstract

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The Event is Supported by: