Instability of some (positive) Einstein metrics under the Ricci flow

Einstein metrics are fixed points (up to scaling) of Hamilton's Ricci flow. A natural question to ask is whether a given metric is stable in the sense that the flow returns to the Einstein metric under a small perturbation. I'll give a brief survey of this area focussing on the case when the Einstein constant is positive. An interesting class of metrics where this question is not completely resolved are the compact symmetric spaces. I'll report on some recent progress with Tommy Murphy and James Waldron where we have been able to use a criterion due to Kroencke to show the Kaehler-Einstein metric on some Grassmannians and the bi-invariant metric on the Lie group G_2 are unstable.