Here are a few things I wrote for myself, but with other people in mind.
- My Spring 2008 lecture in Trivial Notions, entitled "Basic Calculus", on the non-integrability of the Gaussian function by elementary methods. It is based on a paper of Rosenlicht, "Integration in Finite Terms", and a previous lecture of Brian Conrad's.
- Some very short proofs of the basic character theory of representations of finite groups. Quite a bit of this is strongly influenced by the first chapter of Fulton–Harris, "Representation Theory", and one of the proofs is taken from Serre's book "Linear Representations of Finite Groups", of which such notes as these can only be a poor imitation.
- Some very short proofs of the basics of Galois theory, from the definitions to the "fundamental theorem".
- A very short proof of the Bruhat decomposition for GLn(k), for any field k.