Abstract

Many natural and social phenomena adhere to a simple principle: the tendency to minimize energy. Mathematically, this is expressed by variational problems, where one looks for states of equilibrium. When singularities occur (surfaces with irregular points, measures concentrated on thin sets, fractal shapes, etc.) classical tools such as blow up analysis and weak convergence fall short in describing these situations, since they are too rough to capture the fine structures that truly govern the system.

The SingMeas project aims to overcome this limitation by combining two powerful mathematical perspectives. Geometric Measure Theory provides a set of tools to describe and analyze irregular shapes and fragmented objects, while Optimal Transport explains how mass or energy moves in the most efficient way. Together, they open up a new approach for untangling the hidden structures of singularities. The idea is that solutions to complex problems often reflect patterns already present in simpler, singular models. By making those patterns visible, we can better understand the original problem and tackle conjectures that have remained unsolved for decades in areas such as Metric Analysis, Fractal Geometry, Elasticity, and Ergodic Theory. This is achieved by exploring in which direction mass travels from a reference configuration, like the Lebesgue measure, toward the singular structure.

Several popular open problems fall within the scope of the project, including the Ambrosio–Kirchheim flat chain conjecture on the nature of metric currents, Besicovitch’s 1/2-conjecture on the lower density of purely unrectifiable sets, Bouchitté’s vanishing mass conjecture for thin elastic structures, and Furstenberg’s ×2×3 conjecture on the rigidity of measures which are invariant under independent dynamics. 

Besides these theoretical goals, SingMeas also points toward concrete applications. A deeper understanding of singular structures can inspire more realistic models of branched transport systems (like irrigation networks, vascular systems, or supply chains) that must combine efficiency with robustness. 

Duration: 3 years (1/2/2026-31/1/2029)
Host Institution: University of Trento
Budget: 1'502'500,40 €
CUP: E53C25001800001

Research group members:

Postdocs:
Luigi De Masi

PhD  students:
Martina Bellettini
Gianmarco Caldini
Jakub Krukowski

Master students:
Sebastiano Baviello
Amir Nejmi

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