# Rational Elliptic Surfaces and Trigonometry of Non-Euclidean Tetrahedra

*Daniil Rudenko (University of Chicago)*

**06-May-2022, 19:00-20:00 (13 months ago)**

**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.

algebraic geometry

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).

**Stanford algebraic geometry seminar **

**Series comments: **The seminar was online for a significant period of time, but for now is solely in person.
More seminar information (including slides and videos, when available): agstanford.com

Organizer: | Ravi Vakil* |

*contact for this listing |