Standard and helical magnetorotational instability: How singularities create paradoxal phenomena in magnetohydrodynamics

The magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable rotating shear flows, e.g., in accretion discs. What laws of differential rotation are susceptible to the destabilization by the axial or helical magnetic field? The answer to this vital for astrophysical and experimental applications question inevitably leads to the study of spectral and geometrical singularities on the instability threshold of the operator of this problem. The singularities provide a connection between seemingly discontinuous stability criteria and thus explain several paradoxes in the theory of MRI that were kept poorly understood since 1950s.