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".