Daniel Altman, On an arithmetic Sidorenko conjecture, and a question of Alon

Let $G=\mathbb{F}_p^n$. Which systems of linear equations $\Psi$ have the property that amongst all subsets of $G$ of fixed density, random subsets minimise the number of solutions to $\Psi$? This is an arithmetic analogue of a well-known conjecture of Sidorenko in graph theory, which has remained open and of great interest since the 1980s. We will discuss some recent results along these lines, with particular focus on some of the ideas behind a negative answer to a related question of Alon.