  A new point of view about the deep origin of thermodynamic phase transitions
is sketched. The main idea is to link the appearance of phase transitions to
some major topology change of suitable submanifolds of phase space instead of
linking them to non-analyticity, as is usual in the Yang-Lee and in the
Dobrushin-Ruelle-Sinai theories. In the new framework a new possibility appears
to properly define a mathematical counterpart of phase transitions also at
finite number of degrees of freedom. This is of prospective interest to the
study of those systems that challenge the conventional approaches, as is the
case of phase transitions in nuclear clusters.
