The PI proposes to investigate manifolds with integral curvature bounds as opposed to the usual pointwise curvature bounds. Many traditional results carry over, but people do not have a complete picture of what can go wrong with the weaker curvature assumptions. The second and main part of the proposal addresses the specific but very hard question of which 3-sphere bundles over the 4-sphere have positive curvature. Only one of these spaces, namely the 7-sphere, is know to have positive curvature. We expect that several exotic spheres of this type have positive curvature and also that we will get some interesting infinite families of manifolds in dimensions 6 and 7 out of this project.
The PI is proposing to work on two separate projects. The first deals with gleaning information from objects where one has fuzzy rather than specific assumptions about how they curve. The PI has already established some interesting results in this direction, but much is still unknown. The second and more ambitious part is an investigation of how round one can make twisted objects. Here one imagines a sphere as perfectly round and untwisted. There are no intuitive examples in the dimensions that can be visualized, but in dimension 7 there apear to be several objects which are not spheres but still could be fairly round. A positive solution to this problem would solve one very prominent problem and two lesser known problems.
