The theory of currents and the transport equation

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Next lectures
12.11.2025, 14:00 (E2 10 (Leon-Lichtenstein))
19.11.2025, 14:00 (E2 10 (Leon-Lichtenstein))
26.11.2025, 14:00 (E2 10 (Leon-Lichtenstein))
03.12.2025, 14:00 (E2 10 (Leon-Lichtenstein))
10.12.2025, 14:00 (E2 10 (Leon-Lichtenstein))
17.12.2025, 14:00 (E2 10 (Leon-Lichtenstein))
07.01.2026, 14:00 (E2 10 (Leon-Lichtenstein))
14.01.2026, 14:00 (E2 10 (Leon-Lichtenstein))
21.01.2026, 14:00 (E2 10 (Leon-Lichtenstein))
28.01.2026, 14:00 (A3 01 (Sophus-Lie room))
04.02.2026, 14:00 (E2 10 (Leon-Lichtenstein))

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In this course I will introduce the theory of currents in Euclidean spaces, which is nowadays a standard setting to study the Plateau problem in the context of minimal surfaces. Currents represent a notion of generalized oriented manifold, which might have corners and singularities, and enjoy good compactness properties making them useful to study variational problems.

The first part of the course is devoted to the foundations of the theory, and by the end of the course the goal is to introduce the transport equation for currents, which describes their motion through a given velocity field. We will discuss the well-posedness of this PDE in the class of Lipschitz vector fields, and its relation to fluid dynamics and the motion of dislocations in materials.

The lectures will tentatively touch upon the following topics:

• Existence of minimizers for the Plateau problem in the class of integral currents;
• Flat distance;
• Decomposition theorems (Smirnov's theorem; decomposition of integral currents);
• Polyhedral deformation theorem;
• Isoperimetric inequality;
• Transport equation for currents.

The contents can also be adapted or expanded according to the audience.

Several tools and notions from geometric measure theory will be introduced along the way, such as rectifiable sets, Rademacher's theorem, the area formula, and the decomposability bundle.

I intend to keep the course rather informal, giving proofs but sometimes skipping the more technical details, with the idea of giving an overview of the theory. Useful prerequisites are the theory of distributions and basic measure theory. For some background on rectifiable sets you can have a look at section 5 (plus possibly 1,3,4) of my lecture notes for the course "Selected topics in Geometric Measure Theory" that I gave in 2023/24 and that you can find in my homepage, in the teaching section., see the direct link.

Keywords
Currents, Plateau problem, minimal surfaces, transport equation, geometric measure theory

Prerequisites
Basic measure theory. Knowledge of the theory of distributions, differential forms and BV functions is also helpful but not strictly necessary