ICM2014 — Bhargava laudatio

Gowers's Weblog

I ended up writing more than I expected to about Avila. I’ll try not to fall into the same trap with Bhargava, not because there isn’t lots to write about him, but simply because if I keep writing at this length then by the time I get on to some of the talks I’ve been to subsequently I’ll have forgotten about them.

Dick Gross also gave an excellent talk. He began with some of the basic theory of binary quadratic forms over the integers, that is, expressions of the form $latex ax^2+bxy+cy^2$. One assumes that they are primitive (meaning that $latex a$, $latex b$ and $latex c$ don’t have some common factor). The discriminant of a binary quadratic form is the quantity $latex b^2-4ac$. The group SL$latex _2(mathbb{Z})$ then acts on these by a change of basis. For example, if we take the matrix $latex begin{pmatrix}2&15&3end{pmatrix}$, we’ll replace $latex (x,y)$…

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