This talk is concerned with the numerical verification, or otherwise, of a conjecture in spectral geometry due to Kenig, that the spectral radius of the Neumann-Poincaré operator, i.e., the double-layer potential operator in potential theory, is < 1/2 for every bounded Lipschitz domain, equivalently that the same holds for the essential spectral radius.  We study this conjecture in two space dimensions for a class of highly oscillatory piecewise analytic boundaries for which we can compute numerical approximations to the essential spectrum and functionals that determine whether the essential spectral radius is < 1/2 at a continuous level. The tools are a Floquet-Bloch transform, the trapezoidal rule and Nystrom method for analytic functions and associated error estimates, and new (but rather straightforward) Banach space estimates for the spectral radius of an operator in terms of computable quantities for operator approximations. This is joint work with Raffael Hagger (Kiel), Karl-Michael Perfekt (Trondheim), and Jani Virtanen (Reading/Eastern Finland).