Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors

Abstract

We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.