The biggest result in my field in 2010 was the solution to the Erdos-distance problem in the plane by Guth and Katz. This result was quite a breakthrough, and it was a surprise to many. Specifically, they proved the following conjecture of Erdos.
For $E \subset \mathbb{R}^n$ put $\Delta(E) = \lbrace |x - y| : x,y \in E \rbrace$, where $| \cdot |$ denotes Euclidean distance. Then for finite sets $E \subset \mathbb{R}^2$, there exists a universal constant such that
$$ |\Delta(E)| \geq c \frac{|E|}{\log |E|}. $$
