Geometric and Combinatorial Configurations in Model Theory

• Applicants: Professor Dr. Martin Hils; Professor Dr. Amador Martin-Pizarro, since 10/2023; Professorin Dr. Katrin Tent
• Subject Area: Mathematics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 431667816

GeoMod is a Collaborative Research project between France and Germany. Contemporary model theory studies abstract properties of mathematicalstructures 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 them to obtain structural consequences. These may be algebraic or geometric in nature, and can be applied to specific structures such as Berkovich spaces, difference-differential algebraic geometry, additive combinatorics or Erdős geometry.A good example of a configuration implying algebraic structure is the group configuration theorem which asserts that certain combinatorial patterns are necessarily induced by a group, and that moreover the structure of the groups which may give rise to this configuration is highly restricted. This result, which generalizes the coordinatization theorems of geometric algebra was given its definitive form for stable theories by Hrushovski in 1986 and hereafter became one of the most powerful tools in geometric stability, used to resolve open problems in classification theory, in the proof of the trichotomy 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-Breuillard on extensions of the Elekes-Szabó theorem. The model theoretic study of valued fields provides another example of the confluence of stability theory and algebraic model theory. A. 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 replace 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, which has traditionally been very strong both in France and in Germany, but using the sophisticated tools of geometric stability. Finally, we will develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects.

• DFG Programme: Research Grants
• International Connection: France
• Co-Investigator: Professorin Dr. Franziska Jahnke
• Cooperation Partners: Dr. Franck Benoist; Dr. Thomas Blossier; Dr. Elisabeth Bouscaren; Professorin Zoé Chatzidakis, Ph.D.; Dr. Silvain Rideau; Dr. Patricio Simonetta; Professor Dr. Frank Olaf Wagner
• Ehemalige Antragsteller: Dr. Martin Bays, until 9/2023; Dr. Daniel Palacin, until 2/2022