Restriction of eigenfunctions and nodal domains

I will discuss recent progress in the problem of counting nodal domains of eigenfunctions of Laplacian (i.e., counting connected components of the complement to the zero set of  real valued eigenfunctions). This in an old question taken up by Courant and his school. One of many intractable questions is under what conditions one have the number of nodal domains to be unbounded as the eigenvalue goes to infinity. The main difficulty in this problem is that it is known not to be a local property.  
The example I will consider concerns with eigenfunctions of the Laplace-Beltrami operator on compact hyperbolic surfaces. The distinctive property of such a setup is its Quantum (Unique) Ergodicity (to be explained). I will discuss how this could be used in order to deduce strong bounds on eigenfunctions and how this forces the number of nodal domains to grow with the eigenvalue.

(Joint work with J. Bernstein, A. Gosh, P. Sarnak)