The analysis of singular solutions plays an important role in many geometrical and physical problems, even if the problem one is interested in does not directly involve singular solutions,
as singular solutions may appear in the analysis of limits of regular solutions. In this talk, I will first survey a few earlier results involving the analysis of the asymptotic behavior of singular solutions to some conformally invariant equations, of which the Yamabe equation is a prototype. The analysis often has a global aspect and a local aspect, with the former involving the classification of entire solutions, or description of the singular sets, and the latter involving the local asymptotic behavior of the solution upon approaching the singular set. The two aspects are often closely related.  After the brief general survey, I will describe some recent results involving $\sigma_k$ curvature equations.

We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $ S \subset M$, to a metric, $g_\infty$, on $M ∖setminus S$. We prove theorems which describe when $M_j=(M,g_j) $converge in the Gromov-Hausdorff sense to the metric completion, $(M_\infty,d_\infty), of $(M∖setminus S,g_\infty)$. To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains.

The classical isoperimetric theorem
says that the sphere is the least-perimeter way
to enclose given volume. How does the answer
change when the space is given a density that
weights both perimeter and volume? We'll discuss
some recent results following some preliminary
work by undergraduates. The topic is related to
Perelman's proof of the Poincaré Conjecture.

Recently Streets and Tian introduced a geometric flow of almost-Hermitian
structures. We will discuss the motivation for considering such a flow. Moreover, we
will give evidence that the flow reflects the underlying almost-Hermitian structure
of almost complex manifolds.

It is well known that there exist several differentiable or topological obstructions to compact manifolds admitting metrics of positive scalar curvature. On the other hand, the family of manifolds with positive scalar curvature is quite large since any finite connected sum of them is still a manifold admitting a metric of positive scalar curvature. This talk is concerned with the classification question to this family.
         The classical uniformization theorem implies that a two-dimensional compact manifold with positive scalar curvature is diffeomorphic to the sphere or the real projective space. The combination of works of Scheon-Yau and Perelman gives a complete classification to compact three-dimensional manifolds with positive scalar curvature. In this talk we will discuss how to extend Schoen-Yau-Perelman's classification to four-dimension. This is based on the joint works with Bing-Long Chen and Siu-Hung Tang.