[FOM] "localisable" models

Rupert McCallum Rupert.McCallum at acu.edu.au
Fri Oct 9 19:04:47 EDT 2009

The fundamental theorem of projective geometry characterises those
mappings from a projective space to itself (whose range is not
contained in any straight line) which maps straight lines into
straight lines. Jacques Tits showed how this can be generalised to a
result about the special endomorphisms of a structure known as a
"building" which can be associated with any semsimple algebraic group.
Tits obtained the axioms for the theory of buildings by abstracting
from the structure theory of semisimple algebraic groups.
In my thesis I investigate the extent to which these results can be
strengthened to mappings which are defined on subsets of the
underlying space. I obtained results about mappings defined on basic
open subsets of the underlying space in the classical topology (when
the underlying field is a non-discrete topological field) or on sets
of full measure in a basic open set (when the underyling field is a
locally compact Hausdorff topological field). Lately I have decided
that these results are special cases of a general phenomenon that many
topological buildings and topological flag complexes are
"localisable", that is, the theory of the entire structure can be
interpreted in the theory of a basic open subset. I hope to determine
how extensively this phenomenon holds.

I would be interested in hearing if anyone knows of similar results,
where the theory of some topological model is interpretable in the
theory of any basic open subset.

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