Daniil RUDENKO

Daniil RUDENKO  Rational Elliptic Surfaces and Trigonometry of
Non-Euclidean Tetrahedra

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.