Speaker: Pengyu Yang
29 July, 2020 13:30 UTC
Abstract: As a natural generalisation of Dirichlet’s approximation theorem on real numbers, Dirichlet’s approximation theorem on $m\times n$ real matrices tells us the following: given $m$ real linear forms in $n$ variables, we can find an integral vector such that the evaluations of all the linear forms at this integral vector are simultaneous small. In the 1960s Davenport and Schmidt showed that Dirichlet’s theorem is non-improvable for almost all matrices, and they asked if the analogous result holds for a submanifold of the space of $m\times n$ matrices. This problem is related to an equidistribution problem in the space of unimodular lattices in $\mathbb{R}^n$. In this talk I will present some recent progress on this problem, and I will explain its connections to the geometry of Grassmannian manifolds.