  We present a new kind of critical stochastic finite-time-singularity, relying
on the interplay between long-memory and extreme fluctuations. We illustrate it
on the well-established epidemic-type aftershock (ETAS) model for aftershocks,
based solely on the most solidly documented stylized facts of seismicity
(clustering in space and in time and power law Gutenberg-Richter distribution
of earthquake energies). This theory accounts for the main observations (power
law acceleration and discrete scale invariant structure) of critical rupture of
heterogeneous materials, of the largest sequence of starquakes ever attributed
to a neutron star as well as of earthquake sequences.
