Rational Elliptic Surfaces and Trigonometry of Non-Euclidean Tetrahedra
Daniil Rudenko (University of Chicago)
Abstract: I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface "remembers" the trigonometry of the tetrahedron: the length of edges, dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. The construction is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.
Audience: researchers in the topic
Comments: The synchronous discussion for Daniil Rudenko’s talk is taking place not in zoom-chat, but at tinyurl.com/2022-05-06-dr (and will be deleted after ~3-7 days).
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