Ram Band (Math, Technion)
A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition.
An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction - those which are connected to saddle points.
These give rise to Neumann domains (a.k.a Morse-Smale complexes). We define Neumann lines and Neumann domains and present their fundamental topological properties.
These in turn allow to discuss some aspects of counting the number of Neumann domains, giving estimates on their geometry and connecting them to the 'usual' nodal domains.
The talk is based on joint works in progress with David Fajman, Mark Dennis and Alexander Taylor.