# Questions tagged [lattices]

Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])

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### Outer automorphisms of finitely generated linear groups

Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded ...
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### Kissing number lower bound vs. upper bound - precise meanings?

According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$: $$K_L < K < K_U.$$ I ...
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### Leech lattice shortest vector vs other 23 cases and E8 cases

In this paper by Viazovska, she said that: "The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at the lattice points and radius $1/\sqrt{2}$." So I think ...
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### Reference on high dimensional polytopes with E8 symmetry

I'm starting to work with high dimensional polytopes. I am interested in uniform polytopes of 16-dimension and of 8-dimension (especially Elte and Gosset polytopes that have E8 symmetry). ...
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### Are topological theta series (taking values in tmf(N)) of lattices good for anything?

I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...
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### maximizing number of lattice points with bounded diameter

Suppose I have a lattice $L$ that's just $\mathbb{Z}^k$ but scaled in every coordinate by some (potentially different) real numbers. Now I want to construct a finite set of lattice points $S \subset L$...
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### Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?

Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form \begin{equation} \begin{pmatrix} 0 & \Delta \\ -\Delta & 0 \end{pmatrix}, \quad ...
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### K3 surfaces with no −2 curves

I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
1 vote
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### What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid?

This question is a copy of one I asked in the Math StackExchange forum a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most people on the ...
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### Successive minima and the basis of lattice

I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...
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### Minimal number of half-spaces needed to cover a lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and $w\in\mathbb{R}^n-\Lambda$. A set $S\subseteq\Lambda$ of lattice points will be called a set of waypoints for $\Lambda$ from $w$ if it has any of the ...
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### Why should Serre's conjecture on congruence subgroup property hold?

There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$. Whether a lattice in the group satisfies the congruence subgroup property, ...
Let $\mathcal L$ be the space of lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...
### Kissing the Monster, or $196,560$ vs. $196,883$
The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...