Topology and Phase Transitions:. Towards a Proper Mathematical Definition of Finite N Transitions

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.