Title: Neumann Nodal Domains
Consider a Laplacian eigenfunction on a two-dimensional manifold. The eigenfunction zero set (also nodal set) partitions the manifold to connected components called nodal domains. Although nodal domains are being studied for over two centuries, there are numerous open questions concerning the number of nodal domains and their morphology. We propose the study of a new partition of the manifold, which is generated by the gradient flow lines rather than by the zero set of the eigenfunction. The domains of such partition would be called Neumann nodal domains, for a reason to be explained in the talk. We present some first results in the study of Neumann nodal domains and point out further exploration directions and the connection to the ‘usual’ nodal domains. The talk is based on some joint works in progress with David Fajman, Peter Kuchment, Mark Dennis and Alexander Taylor.