How rare are simple Steklov eigenvalues? (joint w/ IP seminar)

Steklov eigenvalues, introduced by Steklov in 1902, are a type of
eigenvalue arising in boundary value problems. In geometric analysis,
there is a deep connection between extremal Steklov eigenvalue
problems and free boundary minimal surface theory. Steklov eigenvalues
can be viewed as eigenvalues of the Dirichlet-to-Neumann operator,
which plays a central role in inverse problems. They arise in fluid
dynamics, influencing the behavior of liquids in containers and
informing the design of engineering structures. In the talk, we will
discuss the question how rare simple Steklov eigenvalues are on
manifolds and its applications in nodal sets and critical points of
eigenfunctions.