# [FOM] geometry and mathematical logic

John Baldwin jbaldwin at uic.edu
Wed Aug 5 09:44:54 EDT 2009

```There has been  a thorough investigation and conceptualisation of the
Borel-Tits results using model theoretic methods.

Let G be an algebraic group over the algebraically closed field K
and H be an algebraic group over the algebraically closed field L.

Borel-Tits: Every pure group isomorphism s between G and H can be factored
as an isomorphism between K and L followed by an \$L\$- quasi-rational
function.

A key lemma is the `Weil-Hrushovski' theorem.  Every construction group is
definably isomorphic to an algebraic group.

Initiated by Zilber, the argument is expounded well in chapter 4 of
Poizat's Groupes Stables.  A not terribly good translation is available as
Vol 87 in the AMS Math Surveys and Monographs series.

These arguments are related to the Cherlin-Zilber conjecture that a simple
group of finite Morley rank is an algebraic group.

On Tue, 4 Aug 2009, Rupert McCallum wrote:

>
> The fundamental theorem of projective geometry states that a bijection from a projective space of dimension greater than one to itself which sends straight lines onto straight lines is a projective transformation possibly composed with an automorphism of the underlying field. This result can be viewed as a consequence of the fact that projective geometry can be interpreted in the theory of fields and vice versa.
>
> Tits showed that the fundamental theorem of projective geometry generalises to a result about the automorphism group of the building of a split semisimple algebraic group G defined over a field k of rank greater than one. The result stated above is a consequence of the special case where G=SL(n,k), n>2. The other cases give information about various other geometries.
>
> Has any work been done relating this more general result to results in mathematical logic?
>
>
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John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science  M/C 249
jbaldwin at uic.edu
312-413-2149
Room 613 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607

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