University of Bristol |

**Title:** On the nodal counts of the Dirichlet-to Neuman eigenfunctions

**Abstract: **The well-known Courant Nodal domain theorem gives an upper bound for the nodal count of a Laplace eigenfunction on a compact manifold. The zero set of an eigenfunction is called the nodal set and its complement is the Nodal domain. The study of nodal sets and nodal domains is a fascinating area in spectral geometry. Courant’s theorem is a classical result in this area. However, almost nothing is known when we consider Dirichlet-to-Neumann eigenfunctions. We discuss how this celebrated theorem of Courant can be extended to the eigenfunctions of the Dirichlet to Neumann operator. This is joint work with David Sher.