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.

I will discuss natural energy functionals related to the
existence of holomorphic structures on vector bundles and show how
inauspicious Hodge data implies blow up of minimizing sequences.
Grassmann embeddings and an analytic perspective on stability in the
sense of Gieseker and Mumford plays an important role.

The classical Kauffman bracket is an invariant of knots in
space. It can be generalized to knots drawn on a surface. I will
discuss surprising properties of these generalized Kauffman
brackets.