|Venue:||Seminario del Departamento de Matemática Aplicada I, E.T.S.I. Informática|
Historically, it is a corollary of Steinitz's theorem (1922) that every combinatorial triangulation of the sphere is geometrically embeddable in (Euclidean) 3-space. Grünbaum’s conjecture (1967) stated that every triangulation on any closed orientable surface would be geometrically embeddable in 3-space, but was disproved by Bokowski and Guedes de Oliveira in 2000.
The term “embedding” means that no two (geometric) faces may intersect in their interior. The term “immersion” means “locally embedding”—that is, two general faces may overlap or intersect along a line that is not an edge of either face, but no two faces with a common vertex may intersect in their interior. The focus of the lecture will be on geometric embeddability and immersability of “pyramidal triangulations”— that is, triangulations having a vertex connected to all other vertices. For pyramidal triangulations, an algorithm has been designed (Sabitov, 2012) to find a simpler form for Sabitov polynomials for the volumes of geometric embeddings of such triangulations.
Pyramidal simplicial orientable polyhedra with arbitrary genus have been constructed as geometrically embedded in 3-space by Sabitov and the speaker. Sabitov has also constructed geometric immersions of pyramidal nonorientable triangulations in 3-space for arbitrary even nonorientable genus. However, for the case of odd nonorientable genus, especially for the case of the projective plane, the pyramidal immersion problem has turned out to be not so easy. The speaker conjectures that no pyramidal triangulation of the projective plane can be simplicially immersed in 3-space.
Possible approaches to the pyramidal projective plane immersion problem will be discussed. The lecture may be interesting both for theoretical as well as for computational geometers and topologists.