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H 1 How many cubes are orientable?

Auteurs : Da Silva, Ilda P. F. (Auteur de la Conférence)
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 :
    05B35 - Matroids, geometric lattices
    52A37 - Other problems of combinatorial convexity
    52C40 - Oriented matroids

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 01/10/2018
      Date de captation : 24/09/2018
      Collection : Research talks ; Combinatorics ; Geometry
      Format : MP4
      Durée : 00:25:37
      Domaine : Combinatorics ; Geometry
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-09-24_Da_Silva.mp4

    Informations sur la rencontre

    Nom de la rencontre : Combinatorial geometries: matroids, oriented matroids and applications / Géométries combinatoires : matroïdes, matroïdes orientés et applications
    Organisateurs 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

    Citation Data

    DOI : 10.24350/CIRM.V.19450403
    Cite 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


    Voir aussi

    Bibliographie

    1. 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

    2. 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

    3. 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

    4. da Silva, I.P.F., & Gioan, I. (2018). Rectangles and hyperplanes of the hypercube. Working paper -

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