In the first part of the talk, using ideas from Ricci flow, we get a Li-Yau type gradient estimate for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$.
In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$. And we obtain various monotonicity formulas of entropy.