Vanessa Hahn (UP)
Andreas Hermann (Potsdam)
Let $M$ be a closed spin manifold of dimension $n\geq 2$. For every Riemannian metric on $M$ we define the spinor bundle on $M$, a complex vector bundle whose sections are called spinors. We also define the Dirac operator, an elliptic differential operator of first order acting on spinors. Explicit computations of the spectrum of the Dirac operator are only possible for particular metrics (e.g. round metrics on spheres, flat metrics on tori).
In this talk I will give an overview on what is known about the spectrum of the Dirac operator for a generic choice of metric. I will describe some results by Dahl and Ammann-Dahl-Humbert about the eigenvalues of the Dirac operator and then explain my own results about zero sets of eigenspinors.