Perelman's Entropy from Colding's Monotonic Volume and other monotony in parabolic equations

 In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed N-space, which becomes Ricci-flat as N→∞, Perelman did not provide a corresponding explanation for the origin of the entropy. In this talk, I'll show  that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's N-space. I'll also mention how the construction of these quantities actually follow a more general scheme applicable to other parabolic non-linear and, even, non-local equations.