Interactions between Hyperbolic Geometry and Number Theory
4 and 5 of July 2019, University of Luxembourg
Thursday the 4th of July
- 9:30-10:30 Ruth Kellerhals (Fribourg, Switzerland): Apéry's constant and hyperbolic volume. I shall discuss various aspects of hyperbolic volume computations ranging from scissors congruence relations, volumes of ideal regular simplices to universal constants such as pi, Catalan's constant and Apéry's constant.
- 10:30-11:00 coffee break
- 11:00-12:00 Herbert Gangl (Durham, UK): Hyperbolic Tessellations: Gems and Jewels.
We devise a method, based on an idea of Epstein and Penner, to produce infinitely many number theoretically defined `ideal' tessellations of hyperbolic space. The building blocks of the latter often exhibit beautiful symmetry, at least after a non-trivial transformation. They even give rise to some enchanting jewellery, the `Gems of Hypolytos', and you will be treated to a hands-on experience.
- 12:15 pizza and salad
- 14:30-15:30 Dan Yasaki (Greensboro, North Carolina): Tessellations of Hyperbolic 3-space and Applications. There are natural tessellations of hyperbolic 3-space by ideal polytopes that arise from the study of perfect Hermitian forms over imaginary quadratic fields. These tessellations can be used to compute interesting aspects of certain arithmetic groups. In this talk, I will present some of the background of these tessellations, some of the computational difficulties that arise, and examples of Bianchi forms and group presentations of general linear groups over imaginary quadratic fields that can be obtained by such tessellations.
- 15:30-16:00 coffee break
- 16:00-17:30 Guided tour through the blast furnaces
- 17:30 barbecue
Friday the 5th of July
- 9:30-10:30 Philippe Elbaz-Vincent (Grenoble, France): Geometry, numbers and cohomology: A perfect story...
- 10:30-11:00 coffee break
- 11:00-12:00 Graham Ellis (Galway, Ireland): Homotopies and cohomology of (arithmetic) groups. The homotopy invariance of homology is a basic tool for calculating the homology of a space. In this talk I'll explain how explicit descriptions of homotopies between spaces can also help with homology calculations. In particular, I'll suggest how such descriptions might be useful in calculating Hecke operators on the cohomology of some arithmetic groups.
- 12:15 Nepalese food
- 14:30-15:30 Rob de Jeu (Amsterdam, The Netherlands): Tessellations of hyperbolic 3-space and Bloch groups. Download abstract from here.
- 15:30 refreshments
All lectures took place in room MNO 1.020 of the Maison du Nombre on the Belval campus of University of Luxembourg.