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How many cubes are orientable?
Multi angle
Authors : Da Silva, Ilda P. F. (Author of the conference)
CIRM (Publisher )
05B35 - Matroids, geometric lattices
52A37 - Other problems of combinatorial convexity
52C40 - Oriented matroids
CIRM (Publisher )
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Abstract :
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
MSC Codes : 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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 01/10/2018 Conference Date : 24/09/2018 Subseries : Research talks arXiv category : Combinatorics ; Metric Geometry Mathematical Area(s) : Combinatorics ; Geometry Format : MP4 (.mp4) - HD Video Time : 00:25:37 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-09-24_Da_Silva.mp4 
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Information on the Event
Event Title : Combinatorial geometries: matroids, oriented matroids and applications / Géométries combinatoires : matroïdes, matroïdes orientés et applicationsEvent Organizers : Gioan, Emeric ; Ramírez Alfonsín, Jorge Luis ; Recski, Andras Dates : 24/09/2018 - 28/09/2018 Event Year : 2018 Event URL : https://conferences.cirm-math.fr/1859.html
Citation Data
DOI : 10.24350/CIRM.V.19450403Cite this video as: 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 |
See Also
- [Multi angle] Generalizations of Crapo's Beta Invariant / Author of the conference Gordon, Gary ; Author of the conference McMahon, Liz.
- [Multi angle] Delta-matroids as subsystems of sequences of Higgs lifts / Author of the conference Bonin, Joseph E..
- [Multi angle] Tverberg-type theorems with altered nerves / Author of the conference De Loera, Jesus A..
- [Multi angle] The concatenation operation for uniform oriented matroids and simplicial (or simple) polytopes / Author of the conference Lawrence, Jim.
- [Multi angle] Approximating clutters with matroids / Author of the conference De Mier, Anna.
- [Multi angle] A matroid extension result / Author of the conference Oxley, James G..
Bibliography
- 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 -
