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Explore the fascinating world of polyhedra with vertices on a quadric in this 50-minute conference talk by Jean-Marc Schlenker at BIMSA. Delve into Steiner's 1832 question about the combinatorial types of such polyhedra and discover the current state of knowledge on this topic. Learn how recent advancements are rooted in the study of ideal polyhedra across various geometries. Examine the characterization of polyhedra inscribed in a sphere using ideal hyperbolic polyhedra properties, as established by Hodgson, Rivin, and Smith. Investigate the latest findings on polyhedra inscribed in one-sheeted hyperboloids or cones, utilizing properties of ideal polyhedra in anti-de Sitter and Half-pipe spaces. Uncover insights into the combinatorial types of polyhedra "weakly inscribed" in two-sheeted hyperboloids, employing an extension of hyperbolic space into de Sitter space. Recognize the ongoing challenge of addressing the same question for one-sheeted hyperboloids. Gain valuable knowledge from this joint work with Jeff Danciger and Sara Maloni, presented as part of the ICBS2024 conference.
