Title: Polynomial effective equidistribution for higher dimensional unipotent subgroups
Abstract: Let $G$ be a semisimple Lie group, $\Gamma$ be a lattice in G and $U$ be a unipotent subgroup of $G$. A celebrated theorem of Ratner says that for any $x \in G/\Gamma$ the orbit $U.x$ is equidistributed in a periodic orbit of some subgroup $U \leq L \leq G$. Establishing a quantitative version of Ratner's theorem has been long sought after. If $U$ is a horospherical subgroup of $G$, the question is well-studied. If $U$ is not a horospherical subgroup, this question is far less understood. Recently, Lindenstrauss, Mohammadi, Wang and Yang established a fully quantitative and effective equidistribution result for orbits of one-parameter (non-horospherical) unipotent groups in some low dimension cases. In this talk, we will discuss a recent equidistribution theorem for unipotent subgroups in higher dimension. Our results in particular provide equidistribution theorems for orbits of the isometry group of a non-degemnrate bilinear form on $\R^n$ in $\mathrm{SL}_n(\mathbb{R})/\Gamma$.