[DMO] John Bamberg: Foundations of hyperbolic geometry

15 May 2023 14:00 till 15:00 - Location: EMMCS Lecture Hall F | Add to my calendar

Abstract: The independent discovery by Lobachevsky and Bolyai of hyperbolic geometry in the 1830's was followed by slow acceptance of the subject from the 1860's on, with the publications of relevant parts of the correspondence of Gauss. A new phase was entered from 1903, when David Hilbert, in his work introducing the "calculus of ends", introduced an axiomatisation for hyperbolic plane geometry by adding a hyperbolic parallel axiom to the axioms for plane absolute geometry. In 1938, Karl Menger (of the famous Vienna Circle) made the important discovery that in hyperbolic geometry the concepts of betweenness and equidistance can be defined in terms of point-line incidence. Since an axiom system obtained by replacing all occurrences of betweenness and equidistance with their definitions in terms of incidence would look highly unnatural, Menger and his students looked for a more natural axiom system. In particular, Helen Skala showed in 1992 that there is a set of axioms whose models are the classical hyperbolic planes over Euclidean fields, and her axioms were the first that contained only first order statements. This talk will be on joint work with Tim Penttila (Emeritus, University of Adelaide) where we endeavour to simplify Skala's axioms and retain a characterisation of the classical hyperbolic planes.