The main aim of the proposal is to study interactions of algebraic geometry with differential geometry and with number theory. First, the proposal aims to explore further the connections between Kaehler--Einstein metrics on Fano orbifolds and Einstein metrics on odd dimensional manifolds that can be realized as orbifold cicrle bundles over algebraic varieties. Another aim is to study the interactions between the geometry of a variety (for instance group actions or Gromov--Witten invariants) and arithmetic problems about the existence of points and rational curves defined over special fields.
If we make a banana or a box out of stretchable material and we start blowing it up, eventually it starts to look like a sphere. This happens because the sphere is the most symmetric way of realizing this shape in space, where the surface tension is most evenly distributed. For arbitrary shapes and in higher dimensions, Einstein wrote down certain conditions for the optimal realizations. These are now called Einstein metrics. Recently (with Boyer and Galicki) I have discovered a new method to contruct such Einstein metrics for many different shapes. A main aim of the proposal is to understand the applicability, scope and limitations of this method.
