Geometric and combinatorial configurations in Model theory

GeoMod AAPG2019 (ANR-DFG)

Model theory studies structures from the point of view of first-order logic. It isolates combinatorial properties of definable sets and uses these to obtain algebraic consequences. A key example is the group configuration theorem which asserts that certain combinatorial patterns are necessarily induced by the existence of a group. This has proven a powerful tool in applications, e.g. in diophantine geometry and more recently to combinatorics.

Valued fields are another example of the confluence of stability theory and algebraic model theory. While algebraically closed valued fields were already studied by Robinson in 1959, the importance of the tools from geometric stability only became apparent in work of Haskell-Hrushovski-Macpherson on elimination of imaginaries, and have in turn triggered the development of the abstract theory of stable domination, leading to applications in non-archimedean geometry.

In the project, we aim to strengthen the recent relations between model theory and combinatorics, develop the model theory of valued fields using tools from geometric stability and carry out an abstract study of the configurations which are a fundamental tool in these two areas.

French coordinator:

Elisabeth Bouscaren (Université Paris-Saclay)

German coordinator:

Martin Hils (Universität Münster)
To the members of the project: Recall to mention the ANR-DFG program GeoMod in your preprints and publications!

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