In 1986, Yau and Peter Li made use of the same methods to study parabolic partial differential equations on Riemannian manifolds. Richard Hamilton generalized their results in certain geometric settings to matrix inequalities. Analogues of the Li-Yau and Hamilton-Li-Yau inequalities are of great importance in the theory of Ricci flow, where Hamilton proved a matrix differential Harnack inequality for the curvature operator of certain Ricci flows, and Grigori Perelman proved a differential Harnack inequality for the solutions of a backwards heat equation coupled with a Ricci flow.