The Edinburgh Lecture Notes on Freiman's Theorem from a minicourse I gave in 2002. That was nearly ten years ago, and I haven't tried to keep these up to date. Nonetheless, they might still be useful. They are heavily influenced by a course Tim Gowers gave in 1999. I lectured on this topic again in MIT in 2005, an undertaking which led to some supplementary notes on Progressions and Convex Geometry. Armed with both of these sets of notes, the reader should have a fairly complete grasp of Chang's bounds for Freiman's theorem.

Some notes on the Bourgain-Katz-Tao sum-product theorem, which I also lectured on at MIT in 2005. Looking at these again, I don't find them to be in very good taste. However they are brief and, at least in some sense, complete. In 2009 I wrote some notes on much the same thing, with an eye to proving the exponential sum estimates of Bourgain, Glibichuk and Konyagin. I got some help from a couple of Fields medallists (Bourgain and Lindenstrauss) in writing these, and I find looking at them again a much more palatable proposition.

At the start of my PhD in 1999 I wrote some notes on Bourgain's bound for 3-term progressions, his work on arithmetic progressions in sumsets and the Heath-Brown-Szemeredi bound for 3-term progressions. This really was a long time ago, and parts of these notes are both embarrassing and extremely out of date in that they do not record recent developments. Nonetheless I know that occasionally people have found the first of these, in particular, to be helpful.

Even longer ago, as an undergraduate in 1998, I entertained notions of becoming a proper number theorist. Around that time I wrote an essay on why is exceptionally close to an integer. I have been asked for a copy of this a few times so I've put it up here for amusement. The essay resurfaced due to this Math Overflow discussion. However it must be admitted that, as Kevin Buzzard points out, I "have just written up a lot of the details of the standard proof that the constant is almost an integer. I think one needs to go a little deeper into the theory to answer the question at hand."

About 5 years ago, when people first started seriously thinking about nonabelian approximate groups, there remained the possiblility that life would make things easier for us by allowing them to have dense models in genuine groups. This example proves that this is not the case. Given what we know now, there is nothing surprising about the example.