Péter Pál Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities

The Alon-Jaeger-Tarsi conjecture states that for any finite field $\mathbb{F}$ of size at least 4 and any nonsingular matrix $M$ over $\mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Mx$ has a 0 component. In joint work with János Nagy we proved this conjecture when the size of the field is sufficiently large, namely, when $61<|\mathbb{F}|\ne 79$. In this talk we will discuss previous results and sketch our proof.