## Archive for August 15th, 2014

### ICM2014 — Bhargava laudatio

August 15, 2014

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 $ax^2+bxy+cy^2$. One assumes that they are primitive (meaning that $a$, $b$ and $c$ don’t have some common factor). The discriminant of a binary quadratic form is the quantity $b^2-4ac$. The group SL$_2(\mathbb{Z})$ then acts on these by a change of basis. For example, if we take the matrix $\begin{pmatrix}2&1\\5&3\end{pmatrix}$, we’ll replace $(x,y)$ by $(2x+y, 5x+3y)$ and end up with the form $a(2x+y)^2+b(2x+y)(5x+3y)+c(5x+3y)^2$, which can be rearranged to
$(4a+10b+25c)x^2+(4a+11b+30c)xy+(a+3b+9c)y^2$
(modulo any mistakes I may have made). Because the matrix is invertible over the integers, the new form can be transformed back to the old one by another change of basis, and hence takes the same set of values. Two such forms are called equivalent.