[FOM] geometry and mathematical logic

[John Baldwin]

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?
>
>
>      __________________________________________________________________________________
> Find local businesses and services in your area with Yahoo!7 Local.
> Get started: http://local.yahoo.com.au
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
> !DSPAM:4a792e03134238830384899!
>
>

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