A search for a good definition of surface leads to the rectifiable currents of geometric measure theory. with interesting advantages and disdvantages.

We present a proof of Bernstein's theorem for minimal surfaces which makes use of major techniques from geometric measure theory.

We present a number of evolution equations which arise in differential geometry starting with the linear heat equation on a Riemannian manifold and proceeding to the curve shortening flow, mean curvature flow and Hamilton's Ricci flow for metrics. We shall first show that a solution of the heat equation on a compact Riemannian manifold converges smoothly to its average value as $t \rightarrow \infty$, using only techniques which carry over to the nonlinear evolution equations presented in the lectures. We will then concentrate mainly on curve shortening and mean curvature flow which exhibit many of the features particular to a variety of nonlinear parabolic equations.

These notes are designed to give a heuristic guide to many of the basic constructions of differential geometry. They are by no means complete; nor are they at all exhaustive. Some of the elementary topics which would be covered by a more complete guide are: geodesics and conjugate points; Lie derivative and the flow of a vector field; coordinate construction techniques; GauB-Bonnet Theorem; Bochner formulae; de Rham cohomology; Lie groups. Despite these and other omissions, I hope that the notes prove useful in motivating the basic geometric constructions on a manifold.

The mathematical motivation for studying vector bundles comes from the example of the tangent bundle $TM$ of a manifold $M$. Recall that the tangent bundle is the union of all the tangent spaces $T_mM$ for every $m$ in $M$. As such it is a collection of vector spaces, one for every point of $M$.

The physical motivation comes from the realisation that the fields in physics may not just be maps $\phi : M \rightarrow C^N$ say, but may take values in different vector spaces at each point. Tensors do this for example. The argument for this is partly quantum mechanics because, if $\phi$ is a wave function on a space-time $M$ say, then what we can know about are expectation values, that is things like: \[\int_M \langle\phi(x), \phi(x)\rangle dx \] and to define these all we need to know is that $\phi(x)$ takes its values in a one-dimensional complex vector space with Hermitian inner product. There is no reason for this to be the same one-dimensional Hermitian vector space here as on Alpha Centuari. Functions like $\phi$, which are generalisations of complex valued functions, are called sections of vector bundles.

We will consider first the simplest theory of vector bundles where the vector space is a one-dimensional complex vector space -line bundles.

Recent years have seen an upsurge of interest by mathematicians in problems arising from the Einstein equations, with many new and surprising results being established using techniques from pde and differential geometry. These three lectures provide an introduction to some of the problems of current interest, in particular those arising from spherical symmetry or motivated by known behaviour of spherically symmetric spacetimes.

This lecture introduces some of the basic properties of hyperbolic equations, illustrated by the specific example of the wave equation in $\mathbf{R}^3$. Emphasis is placed on the geometric nature of many of the constructions, and on describing different approaches to the existence problem.