Speaker: Shahriar Mirzadeh
9 December, 2020 14:30 UTC
Abstract: Consider the set of points in a homogeneous space $G/\Gamma$ whose $g_t$-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of whole space. This conjecture is proved when $G/\Gamma$ is compact or when has real rank. In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: $\mathrm{SL}(m+n, \mathbb{R})/\mathrm{SL}(m+n, \mathbb{Z})$ and $g_t = \mathrm{diag}{e^{t/m}, \dots , e^{t/m}, e^{-t/n}, \dots, e^{-t/n}}$. This homogeneous space has many applications in Diophantine approximation that will be discussed in the talk if time permits. This project is joint work with Dmitry Kleinbock.