We study curvature estimates of Ricci flow on complete 3-dim manifolds
without bounded curvature assumptions. Especially, from a more general
curvature preservation condition, we derived that nonnegative Ricci
curvature is preserved for any complete solution of 3-dim Ricci flow. A local
version of Hamilton-Ivey estimates is also obtained. Using that the nonnegative
Ricci is preserved under any 3-dim Ricci flow complete solution, we can prove the strong uniqueness of the Ricci flow with bounded nonnegative Ricci curvature and uniform injective radius lower bound as initial assumptions. This is joint work with Bing-Long Chen and Zhuhong Zhang.

We will give a mathematically-oriented review about the
geometry of the internal six-dimensional space M_6 in string theory
(with particular attention to the "type II" variety). In particular
we will be interested in vacua which have a property called
"supersymmetry." We will show what kind of constraints this
physical requirement puts on M_6. One reason this is interesting
mathematically is that the conditions we will get are a natural
generalization of the concept of Calabi-Yau manifold.

Abstract: Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last decade or so there have been several scientific proposals to achieve invisibility. We will introduce some of these in a non-technical fashion concentrating on the so-called "transformation optics" that has received the most attention in the scientific literature.

The almost Matthieu operator arises as a model for Bloch electrons in a magnetic field. Aubry and Andre famously made a conjecture about the spectral properties of this operator more than thirty years ago. In the process of its study, variations of the conjecture arose naturally. Although the original conjecture was recently settled, new problems remain unsettled. We discuss some of these open problems and possible methods (and their shortcomings) to their solution.