I also have an active interest in geometry, largely stemming from the geometric structures that arise in string theory, and especially the AdS/CFT correspondence. For example, the two-dimensional surface of a perfectly round sphere has constant positive curvature, and there is a natural generalization of this notion to higher dimensions where they are called Einstein manifolds (the name comes from the close relation to Einstein's equations in general relativity). A special type of odd-dimensional Einstein manifold plays an important role in the AdS/CFT correspondence, and this has been the focus of some of my research. AdS/CFT relates such geometric objects to quantum field theories in three and four spacetime dimensions, where the geometry is effectively recast in terms of representation theory.
More generally I'm interested in any geometric structures that arise in theoretical physics.