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Model theory, pure and applied

Final Activity Report Summary - MODPAP (Model theory, pure and applied)

The project is in the area of model theory (a branch of mathematical logic) and its applications to other parts of mathematics. The objectives of the project were first to find analogues and generalisations of "stability theory" and "stable group theory" for wider classes of first order theories, and secondly to make progress on several problems at the intersection of algebraic geometry and model theory. The project has been extremely successful and Pillay, with collaborators, has made progress on all parts of the project. Moreover in some respects this work has set the agenda for contemporary research in model theory.

Here are some highlights of the research output. The paper "Groups, measures, and NIP" (with Hrushovski and Peterzil), which solved the so-called "Pillay conjecture" for groups in o-minimal structures, was written in 2005-6 and published in Journal AMS in 2008. It has been very influential in the subject. A follow-up paper "On NIP and invariant measures" (with Hrushovski) was written and submitted in 2007 and is currently being revised. Both these papers make serious progress on the first part of the project, especially in replacing types by measures, and studying forking, in first order theories without the independence property.

Substantial progress on the second part of the project has been made in published papers on the arithmetic of differential equations and on the model theory of Kaehler manifolds (with Moosa), and also in work on differential algebra and number theory (with Bertrand and also with Benoist-Bouscaren) which is currently being written up. The work with Bertrand generalises in various ways classical number-theoretic results of Lindemann as well as more recent differential algebraic results of Ax, to commutative algebraic groups over function fields.

Many other research projects have been initiated and pursued, including a stability-theoretic study of the free group on more than one generator, in the light of work of Sela.

Pillay has published around 20 papers, as well as an edited book, during his period as a Marie Curie Chair. During this period he has collaborated with mathematicians from USA, Canada, France, Italy, Spain, Poland, Israel, and Colombia.

During the period as Marie Curie Chair, Pillay has given several courses at Leeds, as well as several "tutorials" at training meetings. Two Ph.D. students of Pillay have graduated since 2005, and currently Pillay has five postgraduate students at Leeds. Pillay has supervised two postdocs in 2007-2008. Also there has been a stream of research visitors and visiting students over the whole period of the project.

Pillay has organised major international conferences at Leeds for each of the years 2006, 2007, 2008. In 2006 it was on o-minimality, in 2007 on differential fields, and in 2008 on generalised stability theory. He has also helped organise other conferences, including the British Logic Colloquium in Oxford 2006, and the MODNET meeting at Luminy, Marseilles, in 2007.