Servicenavigation

Detailseite

Zurück

Projekt Druckansicht

Geometrische und kombinatorische Konfigurationen in der Modelltheorie

Fachliche Zuordnung Mathematik
Förderung Förderung von 2019 bis 2024
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 431667816
 
Erstellungsjahr 2024

Zusammenfassung der Projektergebnisse

1. Understanding combinatorial configurations from the point of view of model theory and geometry. GeoMod is a Collaborative International Research project between France and Germany, in Model Theory, a branch of Mathematical Logic. 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 obtain 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 Erdös 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 was structured around three themes. First we aimed to strengthen the still fairly recent relations between model theory and combinatorics. Secondly, we aimed 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 aimed to develop a more abstract study of the geometric and combinatorial configurations which are a fundamental tool in the previous two subjects, as well as some of their other applications, making this the unifying theme of the programme : Geometric and combinatorial configurations in Model theory Amongst the major results obtained in the project : We will focus here on the first two themes. Within the applications to combinatorics : generalizations of the classical results of Elekes-Szabo on groups to approximate subgroups, the study of amalgamations in additive combinatorics, some new incidence results à la Szemerédi for the characteristic p case. Within the Model theory of valued fields : elimination of imaginaries for new large classes of Henselian fields, and the first realization of one of our tentative long term goals, the first real model-theoretic approach to the tilting correspondence for perfectoid fields . Scientific production : Since the beginning of the programme, four years ago, 37 articles have been published or accepted, and 42 preprints have been deposited in open access archives. Due to the specific conditions of 2020 and 2021, during those two years we organized and participated only to « online » conferences or small online working groups. We have organized in 2023 two major « in person » conferences and our members have participated in numerous conferences, workshops, summer schools where they have given talks about their results obtained in our program. One thematic semester took place (hybrid) at the Fields Institute : Thematic Program on Trends in Pure and Applied Model Theory (July 1 - December 31, 2021), an event many of our members participated in. Factual information The GeoMod Project is a French-German collaboration in pure mathematics. It was coordinated for the French group by Elisabeth Bouscaren (CNRS, Université Paris-Saclay) and for the German group by Martin Hils (Universität Münster). There were two main institutions in France : the University Paris-Saclay and the University of Lyon 1 (coord. Frank Wagner) and two main institutions in Germany, the University of Münster and the University of Freiburg (coord. Amador Martin-Pizarro). The project started on January 1, 2020 and lasted for 48 months. The ANR financed the French nodes with 156 600 euros and the DFG financed the German nodes with 124 000 euros.

Projektbezogene Publikationen (Auswahl)

 
 

Zusatzinformationen

© 2026 DFG Kontakt / Impressum / Datenschutz

Textvergrößerung und Kontrastanpassung