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This note is written for the Festschrift in honor of Professor ChristerKiselman. Multiplier ideal sheaves identify the location and the extent of thefailure of crucial estimates. In this note we will discuss and explain thehistoric evolution of the notion of multiplier ideal sheaves, especially theinterpretation from the viewpoint of destabilizing subsheaves in the context ofterminating or bounding an infinite process. We will also discuss the approachof constructing rational curves in Fano manifolds by using dynamic multiplierideal sheaves and singularity-magnifying complex Monge-Ampere equations. Thisapproach is still under development with details in the process of being workedout. We will indicate where details still need to be worked out.

Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds