# Rational Elliptic Surfaces and trigonometry of Non-Euclidean Tetrahedra

## Geometry Luncheon Seminar

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Dmitri Burago.

**Speaker:** Daniil Rudenko, U 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.

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 10/30/2019

**Time:** 12:00pm - 1:30pm