Title: Numerical characterizations of reductions
Abstract: There are several generalizations of the classical Samuel multiplicity that can be used to characterize the reductions of an arbitrary ideal in a local ring. A result of H. Flenner and M. Manaresi generalizes the multiplicity theorem of Rees (and Böger's generalization) by using the j-multiplicity, an invariant introduced by R. Achilles and M. Manaresi. In the context of joint reductions of ideals of finite colength, I. Swanson proved that the joint reductions are also characterized by the Samuel multiplicity. In this talk we survey these results and discuss ways to extend Swanson's result to arbitrary ideals.