# Expository

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.

### Blow-up of QNLW with Small Initial Data à la Christodoulou

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

### Approximating Continuous Functions

Expository note on the Runge phenomenon

### Notes on Sobolev Spaces

Lecture notes introducing Sobolev spaces.

### Whitney Extension and Counterexample to Sard's Theorem

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

### The non-existence of black holes in (2+1) dimensional general relativity

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

### Mariş's Theorem

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

### One and two (space-time) dimensional general relativity

Life is boring in low dimensions.

### Gravitational red shift and Killing horizons

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

### A Cute Little Wave Estimate

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

### How to ballast your ship or why my kitchen sink is always dirty

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 …