Subelliptic and Magnetic Operators and Their Interactions

Subelliptic operators are differential operators that, although not elliptic, still provide a gain in regularity. They often arise as sums of squares of vector fields satisfying the Hörmander condition, which ensures that their commutators span the whole space and compensate for the lack of ellipticity.

Magnetic Schrödinger operators offer a natural example. While they are classically elliptic, their semiclassical symbol vanishes on a large set, making them resemble subelliptic operators. In this setting, the commutators of the associated operators encode the magnetic field, linking the Hörmander condition to magnetic effects.

Subelliptic operators were extensively studied in the 1970s and have recently regained attention through modern approaches in geometry, harmonic analysis, and spectral theory, with broad applications in physics and engineering. At the same time, Schrödinger and Dirac operators with magnetic fields are central in mathematical physics, with renewed interest driven by applications such as superconductivity and graphene, and by advances on longstanding problems like tunneling.

The program aims to unite researchers from both areas to encourage collaboration and tackle key open problems.