The geometry of stable lattices in Bruhat-Tits buildings | Vidéo | Carmin.tv

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The geometry of stable lattices in Bruhat-Tits buildings

By Mima Stanojkovski

Appears in collection : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codes 2023

Let $K$ be a discretely valued field with ring of integers $R$ and let $d$ be a positive integer. Then the rank $d$ free $R$-submodules of $K^{d}$ (called $R$-lattices) are the $0$-simplices of an infinite simplicial complex called a Bruhat-Tits building. If $O$ is an order in the ring of $d\times d$ matrices over $K$, then the collection of lattices that are also $O$-modules (called $O$-lattices) is a non-empty, bounded and convex subset of the building. Determining what these subsets are is in general a difficult question. I will report on joint work with Yassine El Maazouz, Gabriele Nebe, Marvin Hahn, and Bernd Sturmfels describing the geometric features of the set of $O$-lattices for some particular orders. If time permits, I will also define spherical codes in Bruhat-Tits buildings and show how these fit in this framework and how they give rise to codes of submodules over chain rings.

Information about the video

Date of recording 06/06/2023
Date of publication 28/06/2023
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Jean Petit
Format MP4

Citation data

DOI 10.24350/CIRM.V.20054903
Cite this video Stanojkovski, Mima (06/06/2023). The geometry of stable lattices in Bruhat-Tits buildings. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20054903
URL https://dx.doi.org/10.24350/CIRM.V.20054903

Domain(s)

Bibliography

EL MAAZOUZ, Yassine, HAHN, Marvin Anas, NEBE, Gabriele, et al. Orders and polytropes: matrix algebras from valuations. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, 2021, p. 1-17. - http://dx.doi.org/10.1007/s13366-021-00600-4
EL MAAZOUZ, Yassine, NEBE, Gabriele, et STANOJKOVSKI, Mima. Bolytrope orders. International Journal of Number Theory, 2022. - https://doi.org/10.1142/S1793042123500471
STANOJKOVSKI, Mima. Submodule codes as spherical codes in buildings. Designs, Codes and Cryptography, 2023, p. 1-24. - http://dx.doi.org/10.1007/s10623-023-01207-7

MSC codes

Document(s)

https://www.cirm-math.fr/RepOrga/2889/Slides/Stanojkovski_slides.pdf

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