## Final Activity Report Summary - MODNET (Model theory and applications)

Model theory is a branch of mathematical logic which studies and compares abstract structures arising in mathematics (such the real numbers or the complex numbers) from the viewpoint of what can be said about them in a fixed formal language. The subject has always had strong connections to other areas of mathematics, and there are links with some areas of theoretical computer science. In the past decade, model theory has reached a new maturity, leading to striking applications to number theory, algebra and geometry, as well as strong interactions with areas such as group theory, representation theory of finite-dimensional algebras, and the study of the p-adics.

Amongst the major scientific achievements to which the Research Training Network MODNET has contributed we highlight major progress on the Cherlin-Zilber algebraicity conjecture and a solution to Pillay's conjecture on groups interpretable in o-minimal structures. Both of these provide an illustration of the types of problems addressed by model theory and emphasise its interaction with other parts of mathematics. Both conjectures establish a link between an object defined rather abstractly and generally in model theoretic terms, and a 'classical' object very familiar in the rest of mathematics: such types of results are often hugely significant in producing applications of model theory to other parts of mathematics. The algebraicity conjecture is that a simple group of finite Morley rank (the abstract, model theoretic concept) is an algebraic group over an algebraically closed field (a well understood, classical concept). The programme for attacking this conjecture is to follow in part the proof of the classification of the finite simple groups and major steps in that programme have now been completed. Pillay's conjecture relates groups interpretable in o-minimal structures (abstract, model theoretic concept) to real Lie groups (classical concept). The proof of this has involved the development of tools such as homology, familiar from other parts of mathematics, in a more abstract setting, and also some new, purely model-theoretic tools which are also being studied in their own right.

MODNET was highly successful in producing new developments in all aspects of model theory and delivered high quality training to young researchers in both the sophisticated tools of pure model theory, and in the other areas of mathematics where they are likely to be applied. The network organised four summer schools which provided solid training in the core theory underlying the project, and four training workshops which were more advanced and closer to current research. At the research level, the network organised three research workshops and two major international conferences. These events provided a focus for interactions between young and senior researchers from the partners, the wider international community of model theorists, and researchers from related areas. The network also funded numerous short visits between partners and visits by external experts to contribute to the project's research activities. All of this activity has provided a strong sense of cohesion to model theory in Europe and has increased its visibility and attractiveness to mathematicians working in other fields.

Amongst the major scientific achievements to which the Research Training Network MODNET has contributed we highlight major progress on the Cherlin-Zilber algebraicity conjecture and a solution to Pillay's conjecture on groups interpretable in o-minimal structures. Both of these provide an illustration of the types of problems addressed by model theory and emphasise its interaction with other parts of mathematics. Both conjectures establish a link between an object defined rather abstractly and generally in model theoretic terms, and a 'classical' object very familiar in the rest of mathematics: such types of results are often hugely significant in producing applications of model theory to other parts of mathematics. The algebraicity conjecture is that a simple group of finite Morley rank (the abstract, model theoretic concept) is an algebraic group over an algebraically closed field (a well understood, classical concept). The programme for attacking this conjecture is to follow in part the proof of the classification of the finite simple groups and major steps in that programme have now been completed. Pillay's conjecture relates groups interpretable in o-minimal structures (abstract, model theoretic concept) to real Lie groups (classical concept). The proof of this has involved the development of tools such as homology, familiar from other parts of mathematics, in a more abstract setting, and also some new, purely model-theoretic tools which are also being studied in their own right.

MODNET was highly successful in producing new developments in all aspects of model theory and delivered high quality training to young researchers in both the sophisticated tools of pure model theory, and in the other areas of mathematics where they are likely to be applied. The network organised four summer schools which provided solid training in the core theory underlying the project, and four training workshops which were more advanced and closer to current research. At the research level, the network organised three research workshops and two major international conferences. These events provided a focus for interactions between young and senior researchers from the partners, the wider international community of model theorists, and researchers from related areas. The network also funded numerous short visits between partners and visits by external experts to contribute to the project's research activities. All of this activity has provided a strong sense of cohesion to model theory in Europe and has increased its visibility and attractiveness to mathematicians working in other fields.