Timeline for What can lattices tell us about lattices?
Current License: CC BY-SA 4.0
12 events
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| Jul 19, 2023 at 16:33 | comment | added | Asaf | @Mark, in $\mathbb{R}^{n}$ (and in general, any nilpotent group) any lattice has a canonical rational Malcev basis (c.f. Raghunathans' book), so you need to consider ``essentially'' only $\mathbb{Z}^{n}$. | |
| Jul 19, 2023 at 16:01 | comment | added | Mark Schultz-Wu | what form the various abstract results in the study of general lattices take for euclidean lattices $\Gamma\subseteq\mathbb{R}^n$, and if any say something novel (meaning that there isn't a well-known direct argument for) in this setting. | |
| Jul 19, 2023 at 15:59 | comment | added | Mark Schultz-Wu | cf this. This has had some applications in questions purely regarding the study of Euclidean lattices, for example Dadush's work approximating the covering radius, or Agarwal et. al's work on metric embeddings of the torus. In these settings, it seemed that some abstract techniques from geometry had been discovered to yield better results than "direct" arguments using lattices. As a result, I'm curious | |
| Jul 19, 2023 at 15:55 | comment | added | Mark Schultz-Wu | @Asaf One example of what I am looking for (in a slightly different direction) can be found in the work of JB Bost, see for example this. This rephrases standard facts about Euclidean lattices in more general language. By itself this is not useful for concrete applications with Euclidean lattices (or at least not better than results one can get via direct arguments), but some related work has been --- for example the "Successive Slopes" of this more general framework have some merits when studying Euclidean lattices, | |
| Jul 19, 2023 at 13:37 | comment | added | Moishe Kohan | Check en.wikipedia.org/wiki/Oppenheim_conjecture to see if it is an example of what you are looking for. | |
| Jul 19, 2023 at 9:28 | history | edited | Max Lonysa Muller |
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| Jul 19, 2023 at 5:15 | comment | added | Asaf | @Mark, my form assumes unimodular lattices. The $O(n)$ part of yours assumes rotation invariance, which is generally not needed. Anyhow, first and foremost, this presentation allows one to discuss a "generic lattice" as one have a finite measure. Plainly speaking, "all" lattices in $PSL_{n}(\mathbb{R})$, for $n\geq 3$, would be "essentially" $PSL_{n}(\mathbb{Z})$, so I don't understand which kinds of result you are after. Certainly any equidistribution theorem would tell you that if you start with a given lattice and modify it in a prescribed manner, you get a random lattice. | |
| Jul 19, 2023 at 4:54 | comment | added | Mark Schultz-Wu | @asaf generally people instead describe it as the double coset space $O(n)\backslash \mathsf{GL}_n(\mathbb{R})/\mathsf{SL}_n(\mathbb{Z})$. This viewpoint is key to some of the non-trivial lattice facts I mentioned, namely the definition of Siegel measure and the Siegel integration formula. If you know of other consequences of viewing the space of all lattices of this form (or of your form) I would be interested in hearing them. | |
| Jul 19, 2023 at 1:05 | comment | added | Asaf | Isn't studying lattices in $\mathbb{R}^{n}$ basically the same as studying the space $PSL_{n}(\mathbb{R})/PSL_{n}(\mathbb{Z})$? | |
| Jul 18, 2023 at 21:03 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tags
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| Jul 18, 2023 at 21:02 | comment | added | YCor | I remember that the Benoist-Quint results on random walks in semisimple Lie groups modulo lattices has consequences on random walks on the torus. But I have nothing more precise in mind. | |
| Jul 18, 2023 at 17:09 | history | asked | Mark Schultz-Wu | CC BY-SA 4.0 |