Francesco Lin (Columbia University)

Abstract: The Dirac equation, first introduced in 1928 to describe the quantum properties of the electron, has played a fundamental role in geometry and topology in the past century. In these talks, I will focus on its properties in three dimensions. In the first talk, I will explain how it can be used to give a new proof of the following theorem of Gromov (and in fact a Riemannian refinement of it): on a closed orientable three-manifold equipped with a volume form, there exist three vector fields which are volume-preserving and are linearly independent at every point. In the second talk (based on joint work with M. Lipnowski), I will explain how the spectrum of the Dirac operator can be computed somewhat explicitly in the case of hyperbolic manifolds, and discuss consequences regarding Floer theoretic invariants.