Torsion of differentials on toric varieties

We introduce an invariant for semigroups with cancellation property. When the semigroup equals the set of lattice points in a rational, polyhedral cone, then this invariant describes the torsion of the differential sheaf on the associated toric variety. Finally, as an example, we present the case of two-dimensional cones (corresponding to two-dimensional cyclic quotient singularities).     

Diffusive clustering in an infinite system of hierarchically interacting diffusions

Summary We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).