# Rational Elliptic Surfaces and Trigonometry of Non-Euclidean Tetrahedra

## Location

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.

The synchronous discussion for Daniil Rudenko’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2022-05-06-dr (and will be deleted after ~3-7 days).