  Ultrathin $\rm Bi_2Se_3$-NbN bilayers comprise a simple proximity system of a
topological insulator and an s-wave superconductor for studying gating effects
on topological superconductors. Here we report on 3 nm thick NbN layers of
weakly connected superconducting islands, overlayed with 10 nm thick $\rm
Bi_2Se_3$ film which facilitates enhanced proximity coupling between them.
Resistance versus temperature of the most resistive bilayers shows insulating
behavior but with signs of superconductivity. We measured the magnetoresistance
(MR) of these bilayers versus temperature with and without a magnetic field H
normal to the wafer (MR=[R(H)-R(0)]/\{[R(H)+R(0)]/2\}), and under three
electric gate-fields of 0 and $\pm2$ MV/cm. The MR results showed a complex set
of gate sensitive peaks which extended up to about 30 K. The results are
discussed in terms of vortex physics, and the origin of the different MR peaks
is identified and attributed to flux-flow MR in the isolated NbN islands and
the different proximity regions in the $\rm Bi_2Se_3$ cap-layer. The dominant
MR peak was found to be consistent with enhanced proximity induced
superconductivity in the topological edge currents regions. The high
temperature MR data suggest a possible pseudogap phase or a highly extended
fluctuation regime.
