In this talk, we will discuss locally conformally flat manifolds with finite total curvature.

We prove that for such a manifold, the integral of the $Q$-curvature
equals an integral multiple of a dimensional constant. This
shows a new aspect of the $Q$-curvature on noncompact complete
manifolds. It provides further evidence that $Q$-curvature controls
geometry as the Gaussian curvature does in two dimension on locally conformally flat manifolds

 "Convergence of Manifolds and Metric Spaces with Boundary"  

We study sequences of oriented Riemannian manifolds with boundary

and, more generally, integral current spaces and metric spaces 

with boundary. We prove theorems demonstrating when the Gromov-Hausdorff 

and Sormani-Wenger Intrinsic Flat limits of sequences of such

metric spaces agree.  Then for sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, non collapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary to obtain converging subsequences where both limits coincide

We explore the structure of the singularities of Yang-Mills flow in dimensions n ≥ 4. First we derive a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow at such singular points, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set. This is joint work with Jeffrey Streets

We obtain a local isoperimetric constant estimate for integral Ricci curvature, which enable us to extend several important tools like maximal principle, gradient estimate, heat kernel estimate and $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, including the collapsed case. This is a joint work with X. Dai and Z. Zhang