Geometric and combinatorial configurations in model theory – GeoMod

GeoMod is a Collaborative International Research project between France and Germany.

Contemporary model theory studies abstract properties of mathematical
structures from the point of view of first-order logic. It tries to isolate combinatorial properties of definable sets such as the existence of certain configurations, or of rank functions, and to use these properties to obtains structural consequences. These may be algebraic or geometric in nature, and can be applied to specific structures such as Berkovich geometry, difference-differential algebraic geometry, additive combinatorics or Erdos geometry.

A good example of a combinatorial configuration implying algebraic structure is the group configuration theorem which asserts that certain combinatorial/dimension theoretic patterns are necessarily induced by the existence of a group, and that moreover the structure of the groups which might give rise to this configuration is highly restricted. This result, which itself generalizes the coordinatization theorems of geometric algebra was given its definitive form for stable theories by Hrushovski in his 1986 PhD thesis and hereafter became one of the most powerful tools in geometric stability theory, used to resolve open problems in classification theory, in the proof of the trichotomy theorem for Zariski geometries, and thereby the crucial component of the model theoretic solution of the function field Mordell-Lang and number field Manin-Mumford conjectures. More recently, the group configuration theorem and its avatars have taken center stage in applications to combinatorics, for example in the work of Bays and Breuillard on extensions of the Elekes-Szabó theorem.


The model theoretic study of valued fields provides another example of the confluence of “pure” stability theory and “applied” algebraic model theory. Abraham Robinson identified ACVF, the theory of algebraically closed nontrivially valued fields, as the model companion of the theory of valued fields already in 1959, and for most of the next half century the theory maintained an “applied” character distinct from the stability theory of “pure” model theory. However, in order to describe quotients of definable sets by definable equivalence relations (imaginaries) in valued fields, Haskell, Hrushovski and Macpherson were led to the theory of stable domination and the pure and applied strands merged. The deep connections between these approaches to the theory of valued fields further manifested themselves in the Hrushovski-Loeser approach to nonarchimedian geometry, in which spaces of stably dominated types replaced Berkovich spaces.

Our project is structured around these three themes: First we aim to strengthen the still fairly recent relations between model theory and combinatorics. Secondly, we aim to develop the model theory of valued fields, a subject which has traditionally been very strong both in France and in Germany, but using the sophisticated tools of geometric stability (or neostability). Finally, we will develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects.