Blackholes vs. naked singularities formation in gravitational collapse: analytical and numerical results and perspectives.

In 1969 Roger Penrose proposed the so called Cosmic Censorship
conjecture, namely the idea that all gravitating systems
undergoing complete gravitational collapse - such as very massive
stars - should always form blackholes. In other words, the
conjecture states that all singularities eventually forming will
be safely hidden to far-away observers by a event horizon. From
the mathematical point of view the conjecture can be read off as
a characterization of the geodesic motion (governed by o.d.e.) in
a gravitational field assumed to be a solution of the Einstein
field equations (a system of p.d.e.), with the addition of
reasonable physical assumptions. No proof of the conjecture is
available so far, while several examples of physically sound
systems exhibiting naked singularities have been found. In recent
years, the application of techniques coming from non-linear
o.d.e. analysis allowed to construct a new mathematical framework
for censorship experiments which can be used in problems so far
considered unsolvable, such as the gravitational collapse of
barotropic perfect fluids. Parallel, relevant results obtained by
other groups working in the numerical simulation of gravitational
collapse will also be reviewed.