# 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.

### A Characterization of Geodesics

About the Einstein-Infeld-Hoffmann theorem characterizing geodesics.

### A Better Estimate of Kempner's Series

The Kempner series recently regained some notoriety due to a Saturday Morning Breakfast Cereal comic (the last panel). The observation …

### Bundles and derivatives

Expository article on the geometric notion of derivatives.

### Bessaga's converse to the contraction mapping theorem

A converse to Banach's fixed point theorem, due to Bessaga, is discussed.

### Conformal compactification of type (p,q) pseudo-Euclidean spaces

We explain how to conformally compactify pseudo-Euclidean spaces with $(p,q)$ signature.

### 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.