On the geometry of nodal domains for random eigenfunctions on compact surfaces

A classical result of R. Courant gives an upper bound for the count of nodal domains (connected components of the complement of where a function vanishes) for Dirichlet eigenfunctions on compact planar domains.  This can be generalized to Laplace-Beltrami eigenfunctions on compact surfaces without boundary. When considering random linear combinations of eigenfunctions, one can make this count more precise and pose statistical questions on the geometries appearing amongst the nodal domains: what percentage have one hole? ten holes? what percentage have their boundary being tangent 100 times to a fixed non-zero vector field? The first 20-25 minutes will give a survey on some fundamental results of Nazarov-Sodin, Sarnak-Wigman, and Gayet-Welschinger before presenting some joint works with I. Wigman (King's College London) and Matthew de Courcy-Ireland (École Polytechnique Fédérale de Lausanne) answering these questions in the last 25-30 minutes.