Spectral problems with mixed Dirichlet-Neumann boundary conditions:   isospectrality and beyond
  Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some results, conjectures and problems related to this variation on the isospectral theme. 
Dmitry Jakobson, Michael Levitin, Nikolai Nadirashvili, Iosif   Polterovich