POSTGRAD RESEARCH STUDENTS' SEMINAR ON THURSDAY THE 26TH OF MAY

A generalization of the fundamental theorem of projective geometry

• Speaker: Rupert McCallum (UNSW)
• Date: Thursday, May 26
• Time: 2pm
• Room: Red Centre 3085

The fundamental theorem of projective geometry is a result due to Tits,concerned with mappings of a flag manifold - the quotient of SL(n,R) by
the subgroup of lower triangular matrices in the case n>2 - to itself which preserve certain fibrations of that manifold.

There is a result of Darboux which states that a bijection of the plane to itself that preserves straight lines is an affine map. It is possible
to apply this to give an elementary proof of the fundamental theorem of projective geometry.

I shall show how to obtain a "localized" version of Darboux which deals with injective continuous maps from an open connected subset of the plane
into the plane which preserve collinearity, stating that all such maps are projective transformations. I shall show how to apply this to obtain a "localized" version of the fundamental theorem of projective geometry.

This is original work done with Michael Cowling. Michael Eastwood has independently obtained similar results.

Afternoon tea to follow in common room

Ben Waterhouse
PGRS Seminar Convenor