Effective equidistribution for some one parameter unipotent flows

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of ${\mathrm{SL}}_2(\mathbb{R})$ in arithmetic quotients of ${\mathrm{SL}}_2(\mathbb{C})$ and ${\mathrm{SL}}_2(\mathbb{R})\times {\mathrm{SL}}_2(\mathbb{R})$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.