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To any k-dimensional subspace of 
$\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian 
$\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and 
$n-k$, respectively. These lattices originate by intersecting the k-dimensional subspace and its orthogonal with the lattice 
$\mathbb {Z}^n$. Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when 
$(k,n) \neq (2,4)$.
An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group 
$\text{PSL}\left( 2,\,\mathbb{Z} \right)$ of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface 
$M$ the length of a systole, which is a shortest closed geodesic of $M$.
