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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.

In this talk I will describe a new link "invariant" (with certain wall-crossing properties) for links L in a three-manifold M, where M takes the form of a surface times the real line. This link "invariant" is constructed via a map, called the q-nonabelianization map, from the
gl(N) skein algebra of M to the gl(1) skein algebra of a covering three-manifold M'. In the special case of M=R^3, this map computes well-known link invariants in a new way. As a physical application, the q-nonabelianization map computes protected spin character counting BPS ground states with spin for line defects in 4d N=2 theories of class-S. I will also mention possible extension to more general three-manifolds, as well as further physical applications to class-S theories. This talk is based on joint work with Andrew Neitzke, and ongoing work with Gregory Moore and Andrew Neitzke.