SFB 878 A04 - Modelltheorie, stabile Gruppen und die entsprechenden Komplexe und Geometrien

Grunddaten zu diesem Projekt

Beschreibung

The combination of methods from algebra and model theory has proven particularly fruitful in recent years in the investigation of algebraic groups and more generally questions from algebraic geometry. In this project we study non-linear actions of algebraic and other, abstractly given groups on combinatorial structures, like buildings or algebraic varieties, which in our context take the form of definable subsets of a structure. The questions addressed here concern characterizations of groups acting on natural geometries associated to algebraic groups, like Tits buildings. Model theory has a natural interest in these questions as one is interested in finding characterizations from which one can deduce that an abstractly given group is (isomorphic to) an algebraic group. This yields information about the first-order structures involved which can then be used for questions concerning algebraic groups and geometry, like the questions about the definable subsets in valued fields. On the other hand, model theory provides tools to construct counter examples, thus often showing that certain characterizations are optimal. Tits and Bruhat-Tits buildings can be described also in a model theoretic way which opens a new approach using recent deep results on valued fields by Hrushovski and Kazhdan or on fields with automorphisms. We plan to investigate combinatorial properties of groups and buildings which can be captured also in model theoretic terms. We propose to investigate a number of questions in this area generally concerning the interactions between groups, geometries and model theory. Tits' center conjecture Weak Moufang conditions Moufang sets and asymptotic group theory Group cohomology and buildings Model theory of arc spaces