  In recent years, there has seen much interest and increased research
activities on Perelman's paper. Section one and two of this paper aim to
establish Perelman's local non-collapsing result for the Ricci flow. This will
provide a positive lower bound on the injectivity radius for the Ricci flow
under blow-up analysis. We also discuss the gradient flow formalism of the
Ricci flow and Perelman's motivation from physics. In this paper, we survey
some of the recent progress on Perelman's functional, reduced volume and
reduced length.
