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.

We present several geometric interpretations of a certain family of
solutions of an "integrable" nonlinear pde. This sheds light on a diverse
range of topics, from the classical Painleve equations to the
quantum cohomology of Fano manifolds.

It is known that many simply connected, smooth topological
4-manifolds admit infinitely many exotic smooth structures. The
smaller the Euler characteristic, the harder it is to construct
exotic smooth structure. In this talk, we construct exotic smooth
structures on small 4-manifolds such as CP^2#k(-CP^2) for k = 2, 3,
4, 5 and 3CP^2#l(-CP^2) for l = 4, 5, 6, 7. We will also discuss the
interesting applications to the geography of minimal symplectic
4-manifolds.

The L^2 norm of the Riemannian curvature tensor is a natural intrinsic analogue of the Yang-Mills energy in purely Riemannian geometry. To understand the structure of this functional, it is natural to consider the gradient flow. I will give an overview of the analytic theory behind this flow, and discuss some long time existence results in low dimensions. Finally I will mention some natural conjectures for this flow and their consequences.