I will try to describe some of the thingswe have learnt, over the past few years, about three fundamental questions inthe global theory of free boundary minimal surfaces; given that context, I willthen focus on one of the latest advances, that is a methodology to gaineffective estimates on the spectrum of Schrödinger-type operators by designingsuitable partitions of the domain and associated collections of mixed boundaryvalue problems. As an application, this allowed us to obtain the firsttwo-sided, explicit affine bounds on the Morse index of various families ofFBMS in the three-dimensional Euclidean ball. The lecture is aimed at thegeneral mathematical audience.