English
How many cubes are orientable?
Multi angle
Auteurs : Da Silva, Ilda P. F. (Auteur de la Conférence)
CIRM (Editeur )
05B35 - Matroids, geometric lattices
52A37 - Other problems of combinatorial convexity
52C40 - Oriented matroids
CIRM (Editeur )
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Résumé :
A cube is a matroid over $C^n=\{-1,+1\}^n$ that contains as circuits the usual rectangles of the real affine cube packed in such a way that the usual facets and skew-facets are hyperplanes of the matroid.
How many cubes are orientable? So far, only one: the oriented real affine cube. We review the results obtained so far concerning this question. They follow two directions:
1) Identification of general obstructions to orientability in this class. (da Silva, EJC 30 (8), 2009, 1825-1832).
2) (work in collaboration with E. Gioan) Identification of algebraic and geometric properties of recursive families of non-negative integer vectors defining hyperplanes of the real affine cube and the analysis of this question and of las Vergnas cube conjecture in small dimensions.
Keywords : cube; matroid; Euclidean space; hyperplanes; oriented matroids
Codes MSC : How many cubes are orientable? So far, only one: the oriented real affine cube. We review the results obtained so far concerning this question. They follow two directions:
1) Identification of general obstructions to orientability in this class. (da Silva, EJC 30 (8), 2009, 1825-1832).
2) (work in collaboration with E. Gioan) Identification of algebraic and geometric properties of recursive families of non-negative integer vectors defining hyperplanes of the real affine cube and the analysis of this question and of las Vergnas cube conjecture in small dimensions.
Keywords : cube; matroid; Euclidean space; hyperplanes; oriented matroids
05B35 - Matroids, geometric lattices
52A37 - Other problems of combinatorial convexity
52C40 - Oriented matroids
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 01/10/2018 Date de captation : 24/09/2018 Sous collection : Research talks arXiv category : Combinatorics ; Metric Geometry Domaine : Combinatorics ; Geometry Format : MP4 (.mp4) - HD Durée : 00:25:37 Audience : Researchers Download : https://videos.cirm-math.fr/2018-09-24_Da_Silva.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Combinatorial geometries: matroids, oriented matroids and applications / Géométries combinatoires : matroïdes, matroïdes orientés et applicationsOrganisateurs de la rencontre : Gioan, Emeric ; Ramírez Alfonsín, Jorge Luis ; Recski, Andras Dates : 24/09/2018 - 28/09/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1859.html
Données de citation
DOI : 10.24350/CIRM.V.19450403Citer cette vidéo: Da Silva, Ilda P. F. (2018). How many cubes are orientable?. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19450403 URI : http://dx.doi.org/10.24350/CIRM.V.19450403 |
Voir aussi
- [Multi angle] Generalizations of Crapo's Beta Invariant / Auteur de la Conférence Gordon, Gary ; Auteur de la Conférence McMahon, Liz.
- [Multi angle] Delta-matroids as subsystems of sequences of Higgs lifts / Auteur de la Conférence Bonin, Joseph E..
- [Multi angle] Tverberg-type theorems with altered nerves / Auteur de la Conférence De Loera, Jesus A..
- [Multi angle] The concatenation operation for uniform oriented matroids and simplicial (or simple) polytopes / Auteur de la Conférence Lawrence, Jim.
- [Multi angle] Approximating clutters with matroids / Auteur de la Conférence De Mier, Anna.
- [Multi angle] A matroid extension result / Auteur de la Conférence Oxley, James G..
Bibliographie
- da Silva, I.P.F. (2005). Recursivity and geometry of the hypercube. Linear Algebra and its Applications, 397, 223-233 - https://doi.org/10.1016/j.laa.2004.10.016
- da Silva, I.P.F. (2008). Cubes and orientability. Discrete Mathematics, 308(16), 3574-3585 - https://doi.org/10.1016/j.disc.2007.07.043
- da Silva, I.P.F. (2009). On minimal non-orientable matroids with $2n$ elements and rank $n$. European Journal of Combinatorics, 30(8), 1825-1832 - https://doi.org/10.1016/j.ejc.2008.12.003
- da Silva, I.P.F., & Gioan, I. (2018). Rectangles and hyperplanes of the hypercube. Working paper -
