Topology Seminar (Main Talk): Non-Euclidean tetrahedra, mixed Hodge structures and rational elliptic surfaces

Seminar | May 8 | 4:10-5 p.m. | 3 Evans Hall

 Daniil Rudenko, University of Chicago

 Department of Mathematics

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.