Optimal criteria for magnetic fields that bind electrons

This talk will provide a basic introduction to the three dimensional Dirac equation that describes an electron interacting with a magnetic field. Over the years a lot of work has gone into constructing zero energy solutions, also known as zero modes, for said equation.
In this talk I will explain the importance of zero modes e.g. I will show how they relate to the stability of the hydrogen atom. After presenting explicit examples, I will state necessary conditions for the magnetic field so that zero modes exist. Here, of particular interest is a sharp inequality that is optimized by a magnetic field whose field lines are interlinking circles. This inequality relates to sharp inequalities for spinors of which some more examples will be given.