The nodal count mystery.
Autor: Peter Kuchment Dizengof
Nodal patterns of oscillating membranes have been known for hundreds of years and are often demonstrated in undergraduate physics classes. They are usually called Chladni figures, although Robert Hooke demonstrated them at Royal Society two centuries before Chladni. Mathematically speaking, these figures are the nodal sets of eigenfunctions of the Laplace operator with Dirichlet boundary conditions on the corresponding domain (or manifold). In spite of them being known for quite a long time, the understanding of these patterns (e.g., how large the nodal set is, or how many nodal domains the pattern splits the membrane into) remains very incomplete. These patterns nowadays attract attention of leading mathematicians and physicists alike. The talk will provide a brief history of the subject and some recent results on the nodal domain count.