# Algebraic Geometry Seminar

#### Classical trigonometry of non-Euclidean tetrahedra

**Speaker:**
Daniil Rudenko

**Location:**
Warren Weaver Hall 317

**Date:**
Tuesday, February 12, 2019, 3:30 p.m.

**Synopsis:**

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.