Equidistribution and freeness on Grassmannians

Equidistribution of primitive lattices in $\mathbb{R}^n$

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive $d$-lattices themselves, we also …

有理子空间及其形状的等分布

Equidistribution of rational subspaces and their shapes

本文利用幂单动力学证明了在特定同余条件下，$\mathbb Q^n$中有理子空间、Grassmannian流形中的点及相关格形状的同时等分布。

正特征下部分格点的联合有效等分布

Joint effective equidistribution of partial lattices in positive characteristic

本文在正特征下研究部分格点的联合有效等分布，利用改进的LU分解和计数技术，证明了格点在不同空间中的等分布性质。

Planes in four space and four associated CM points

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement and the second two essentially arise from the accidental local isomorphism between SO(4) and SO(3)xSO(3). As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings we prove simultaneous equidistribution of all of these objects under two splitting conditions.