Nexus Lecture: Spaces of Geometric Structures

Date: Thursday 16 March 2023

Abstract

The real hyperbolic plane is the first non-euclidean geometry, some of its main features were discovered by G. Saccheri in 1733.

In the 19th century Riemann recognized that a closed surface S with g (at least 2) holes has a 6g-6 dimensional space T(S) of hyperbolic structures, that is, geometric structures on S locally modelled on the real hyperbolic plane. There is then an algebraic model for T(S) as a connected component of an object called character variety, where the component is picked out by a topological invariant, as was understood by W. Goldman in 1980.

The character variety is an object depending on the fundamental group of S and the group SL(2,R). What happens if one replaces the field R by an ordered field F containing infinitely large elements? And what geometric objects does the corresponding component in the character variety built with SL(2,F) parametrize?

In this talk, I will introduce all the objects mentioned above and give an answer to these questions following some ideas of G. Brumfiel.

This will lead us to hyperbolic geometry over the field F, the Tarski principle, valuations, and measured laminations on S. 

About the speaker

Marc Burger was appointed as a full professor to the Mathematics Department at the Swiss Federal Institute of Technology (ETH) in Zurich in 1997. In 1999 he was appointed as Director of the Forschungsinstitut für Mathematik (FIM) at the ETH; he held this position until 2009. He has been a research council member of the Swiss National Science Foundation since 2005.

Marc Burger's main research interest focuses on the study of discrete subgroups of Lie groups, in the broadest sense.

Since 2012 he has been a Fellow of the American Mathematical Society and member of the Leopoldina.

About the Nexus Lecture Series

The Nexus lectures (from the Latin word to bind together) have been established by the School to promote outstanding research in fundamental mathematics and to further future collaborations across different mathematical fields. These lectures will be held every few months and are open to anyone in the UNSW community, as well as the general public.