Georgia Institute of Technology College of Sciences

The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality. 

Rather than going through the proof in detail, I will spend most of the talk introducing the problem and explaining the connection to averaging operators over curves, and explaining why these operators are Fourier integral operators satisfying Sogge's cinematic curvature condition. This condition was originally introduced by Sogge to generalise Bourgain's circular maximal theorem, but it turns out to have useful applications to projection theory.