Ancient solutions to the Ricci flow occur in the singularity analysis of such flows as well as in the Renormalization Group flow in theoretical physics. Explicit examples of such flows are few in the literature. The general aim is to construct such solutions which have interesting geometric properties. In low dimensions, ancient solutions are shown to be rare and indeed can be classified. Our aim is also to explore how common such solutions are in dimensions greater than 4.
Impact
Thus far we have constructed type I ancient solutions including both collapsed and non-collapsed ones on compact torus bundles over a product of Fano Kahler-Einstein manifolds. These occur in all dimensions greater than 7 and generally have positive Ricci curvature. The manifolds include infinitely many homotopy types in each fixed odd dimension, can be of arbitrary large cohomogeneity, and include manifolds which are homeomorphic but not diffeomorphic.
