In this expository talk, we discuss some inequalities holding
for certain solutions of Ricci flow. Ricci flow is a form of the heat
equation for Riemannian metrics. So techniques from the study of the
heat equation apply. Examples of basic inequalities include the
Li-Yau inequality for positive solutions of the heat equation, which
motivated the Harnack inequalities of Hamilton and Perelman for Ricci
flow. Fundamental inequalities of Perelman are for the entropy and for
the reduced volume. Moreover, there are many other inequalities which
hold for certain classes of solutions, such as those proved by
Hamilton, Perelman, and others.