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A new point of view about the deep origin of thermodynamic phase transitionsis sketched. The main idea is to link the appearance of phase transitions tosome major topology change of suitable submanifolds of phase space instead oflinking them to non-analyticity, as is usual in the Yang-Lee and in theDobrushin-Ruelle-Sinai theories. In the new framework a new possibility appearsto properly define a mathematical counterpart of phase transitions also atfinite number of degrees of freedom. This is of prospective interest to thestudy of those systems that challenge the conventional approaches, as is thecase of phase transitions in nuclear clusters.Comment: 6 pages, 1 figur

TOPOLOGY AND PHASE TRANSITIONS: TOWARDS A PROPER MATHEMATICAL DEFINITION OF FINITE N TRANSITIONS