In Donagi-Pantev's programme to deduce geometric Langlands from its abelianised version, wobbly and shaky bundles play a key role. A conjecture by both authors establishes the equality of these loci.

In this talk, I will explain a joint result with Christian Pauly giving a geometric criterion for wobbliness, and report on ongoing work about the applicability of the ideas therein towards the proof of the conjectural equality.