Rigidity of Length Functions over Strata of Flat Metrics

In this thesis we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-LeiningerRafi. Specifically, for any stratum with more unmarked zeroes than the genus, the Σ-length-spectrum of a set of simple closed curves Σ determines the flat metric in the stratum if and only if Σ is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the Σ-length-spectrum of a finite set of closed curves Σ.