The proposer is interested in a long time behaviour of different parabolic flows, such as the Ricci flow, the Yamabe flow and different curvature flows of hypersurfaces in the euclidean space. More precisely, the proposer would like to understand the structure of possible singular limiting metrics one gets. Since the ancient solutions occur as singularity models of finite time singularities, the proposer suggests to study the properties and the classification of those in the case of different flows. One special case of ancient solutions are the gradient shrinking solitons. There is much to be understood about their geometric properties especially in the complete higher dimensional cases which can help the classification of those. Related to the singularities I the proposer also suggests studying the optimal conditions under which one can guarantee the existence of a smooth solution to e.g. the Ricci flow and the mean curvature flow.
The proposer is interested in studying different parabolic geometric flows since their parabolic properties tend to improve the properties of the initial geometric objects. For example, under certain conditions on the initial metric the Ricci flow tends to exist forever and converges to a metric of constant sectional curvature which tells us a lot about the topology of our manifold. That means one can sometimes use the parabolic geometric flows in order to resolve some issues in other mathematical fields. Ancient solutions are the solutions that come from all the way from negative infinity. The physicists are interested in understanding those solutions to the Ricci flow.