4:00 p.m., Monday (October 17, 2005)
University of New South Wales
Mappings that preserve families of curves
This talk is a survey of some results in geometry from Darboux to Tits. First,
any bijection of the plane which maps lines to lines is affine, i.e., a composition
of a linear map and a translation. There are local versions of this result.
Next we consider maps of three-dimensional space which preserve two families of
lines, one family of lines parallel to the y axis and the other family of lines
lying in planes parallel to the xz plane and with gradients equal to the y
coordinates. It is shown that these maps are affine, and that this implies
(for SL(3,R)) a theorem of Tits that the morphisms of a spherical building come
from the group.
A local version and extensions of this result to parabolic geometries are outlined.
Refreshments will be served at 3:45 p.m. (MATX 1115, Math Lounge).