Diff. Geom, Math. Phys., PDE Seminar: Nassif Ghoussoub

I consider two different approaches for breaking scale invariance and restoring compactness for borderline variational problems involving the Hardy-Schrodinger operator -\Delta -\frac{\gamma}{|x|^2} on a domain containing the singularity 0, either in its interior or on its boundary. One consists of adding a linear perturbation, another exploits the geometry of the domain. I discuss the role of various ``positive singular mass theorems" that help account for the critical dimensions below which these approaches fail. This is a joint project with Frederic Robert.