I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 1) in mathematics 2) outside of mathematics, besides …

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice. A set of integer …

This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin: http://mathoverflow.net/questions/85624/fitting-a-mesh-to-a-density-function …

$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 …

Let $\Lambda$ be a lattice (discrete additive subgroup) in $\mathbb R^n$ ($n\geq 2$). In my problem, $\Lambda$ lies in a $k$ dimensional ($1< k\leq n$) subspace of $\mathbb R^n$ …

White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem: If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which conta …

I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addi …

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). …

Here I am only interested in globally irreducible lattices over $\mathbb{Z}$. The basic theorem concerning these says that a globally irreducible lattice is similar to a lattice …

I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be t …

I have a quadratic form $Q(u) = \langle Du , u \rangle = 0$, where $D$ is circulant-symmetric from $\mathbb{R}^{n \times n}$ $D$ has all entries $0$ or $1$ except the diagonal whic …

The $4$-dimensional lattice $\mathbb{Z}^{4}$ has vectors of length $\sqrt{n}$ for any positive integer $n$ by the Four Squares Theorem, but this need not be true for higher-dimensi …

Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function $$L\longmapsto -\log(\hbox{packing density of }L)/ \log(\hbox{covering density of }L)$$ is boun …