  New simulations demonstrate that low-mode, nonradial hydrodynamic
instabilities of the accretion shock help starting hot-bubble convection in
supernovae and thus support explosions by the neutrino-heating mechanism. The
prevailing conditions depend on the high-density equation of state which
governs stellar core collapse, core bounce, and neutron star formation. Tests
of this sensitivity to nuclear physics variations are shown for spherically
symmetric models. Implications of current explosion models for r-process
nucleosynthesis are addressed.
