This is a collection some of the expository notes that I have written, they range in topic, some closer to my research and some farther away. As opposed to posts in "Mathematical Diversions", posts here are intended to be read by other people. I welcome comments and corrections.

Lecture slides that Gustav Holzegel and I used for the January 13-14, 2014 OXPDE Workshop at Oxford University.

Expository note on the Runge phenomenon

Lecture notes introducing Sobolev spaces.

In a first course in differential geometry/topology we are often taught the following version of Sard's theorem: Theorem   …

We show that trapped surfaces cannot exist in (2+1) dimensional relativity.

We describe a theorem due to Mihai Mariş concerning the symmetries of ground states.

Life is boring in low dimensions.

The relationship between gravitational red shift and the surface gravity of black holes is explained.

Let us consider first the linear wave equation \[ \Box u = 0\] on (1+d)-dimensional Minkowski space. A well known property of the wave …

Recently I've been reading a bit of fluid dynamics, and came across the classical problem of free and forced vortices. In this post I …