### Leeds Pure Postgraduate seminar

**Abstract** - Pietro dello Stritto, 19 April 2007

"Inspired by the work of Chatzidakis, van den Dries and Macintyre,
we introduce the concept of a MEASURABLE STRUCTURE, an infinite
structure whose definable sets are equipped with a dimension
and a measure satisfying certain natural axioms. For instance, pseudofinite
fields, namely ultraproducts of finite fields, are important examples.

In this talk, we will mainly be concerned with measurable groups
and fields. We will start by reviewing some preliminary results.
We will not give any proof, but just mention some nice results obtained by
Macpherson and Steinhorn which remain still open in a larger context
called SUPERSIMPLE THEORIES, where measurables theories (theories of
measurable structures) are located:
1. Under reasonably "mild" assumptions measurable
groups have infinite abelian subgroups;
2. An infinite measurable field is quasifinite, perfect and has surjective
maps for every finite Galois extension.

The properties of an infinite measurable field showed in the result 2
above motivate the following conjecture:

FIELD CONJECTURE: An infinite measurable field is pseudofinite.

We expect that an affirmative answer to this conjecture
will carry out the full classification of INFINITE MEASURABLE MOUFANG
POLYGONS as those inherited from the finite cases over pseudofinite
fields. However, there are still hopes to show the classification without
using the field conjecture. One motivation behind this attempt of
classification is the following conjecture analogues of which are known
for various
classes of groups (work of Tits; Kramer, Tent, van
Maldeghem):

GROUP CONJECTURE: If G is an infinite simple
measurable group with a spherical BN-pair of Tits
rank at least 2 then G is isomorphic to a simple
Chevalley group or a twisted analogue over either
a pseudofinite or a difference field (a particular field with an
automorphism).

Finally, we will expose
recent advances towards the classification of measurable
Moufang polygons. We will also mention the most important tool, STRONG
BI-INTERPRETATION (introduced by Ryten - a former PhD student od
Macpherson - in his thesis in Leeds), which should help and lead us,
hopefully, to a proof of the Group Conjecture.
Time permitting, we could talk about the surprising interplay with
octogons and ultraproducts of Frobenius maps".