A general framework on singularity

The minimal model program has led to the introduction of
several notions of singularities. Because of the very nature of the
program, these notions are somewhat of a cohomological type, as they
require certain Cartier conditions to hold. One can however extend
the theory to a completely general setting, by working with Weil
divisors which are homological objects, and many of the features of
the theory extends to such a setting. Already the simple case of cone
singularities offers an interesting class of examples, and by working
in such generality one can also gain some new insight in global
geometry. This talk is based on joint works with C. Hacon, and with S.
Boucksom and C. Favre.