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A Study of the Notion of Ampleness in Model Theory and Tits Buildings

Periodic Reporting for period 1 - AMPLE (A Study of the Notion of Ampleness in Model Theory and Tits Buildings)

Reporting period: 2019-09-01 to 2021-08-31

The project allocates in the area of Model Theory, which forms part of Mathematical Logic and has ample connections to surrounding fields of pure mathematics. The leading aim of the project was to establish new links between the area of Model Theory with Combinatorics and Algebra through a thorough study of the notions of Stationary Independence and Ampleness and use its implications to approach long standing open conjectures in the area, as well as to establish a novel interaction between Model Theory and Geometric Group Theory, or more precisely, the area of Tits Buildings. This is based on findings from the PhD thesis of the ER, where in joint work with her supervisor she was able to use methods from the area around Tits Buildings to resolve a long-standing open conjecture in Model Theory. Conversely, several important notions within buildings reflect abstract and well-studied objects in Model Theory. It hence seems that there is a strong connection between these two fields and exhibiting this clearly will change the area of Model Theory by adding a completely new field of research to it.
During this project the ER explored in weekly meetings with the Supervisor the prospects and limitations of studying Tits buildings as first order structures. Together they introduced a novel structure on the new geometry constructed by the ER in her thesis using the concepts from the paper Trivial Stable Structures with Non-Trivial Reducts by the Supervisor, which yields promise to view the new 2-ample structure as a reduct of a one-based one, the existence of which is still unclear. Further, the ER in collaboration with Prof. Macpherson and Dr. Siniora was able to adapt the notion of stationary independence in order to show that the well-studied bowtie free graph has a simple automorphism group. In an ongoing continuation of that project, the ER with her collaborators is currently combining tools from Geometric Group Theory and Shelah’s paper on Existentially Closed Locally Finite Groups in order to define a stationary independence relation on Philipp Hall’s Universal Group which could allow to decide upon the much studied simplicity of the group of its outer automorphisms. Together with a fellow researcher from Imperial College, Dr. Ghadernezhad, the ER investigated the paper on Simplicity of the Automorphism Groups of Order and Tournament Expansions of Homogeneous Structures by Calderoni, Kwiatkowska and Tent. Consequently, the ER and Ghadernezhad suggest and study various generalisations of the presented results. Finally, the ER together with the master student Shahar Oriel were able to use Hrushovskis famous strictly stable pseudo plane in order to solve an open conjecture. This plane is the only known example of a stable, omega-categorical structure, which is not superstable. As such, it is of great importance in the study of stable structures. Beforehand, the only known proof of its non-superstability relied on deep results by Cherlin, Harrington and Lachlan. Through the development of a novel construction algorithm, the ER and Oriel were not only able to give the first direct proof that the pseudo plane is strictly stable, but also answer negatively a conjecture by Krupinski on small Polish Structures.
Besides the pure research activity, the ER has developed and lectured a course on Geometric Group Theory, which is fundamental as a base to both Tits Buildings as well as the Model Theory of the Free group and will serve as the base to introduce this course in her current institution.
The Model Theory of Tits Buildings remains a promising new area and the project has advanced the understanding of limitations and perspectives of this interaction. The ER was able to obtain a tenure track position as Assistant Professor at the American University of Cairo, where she can continue the development of the area long term and build a group around it. In particular, she will use the material developed during the project to establish a course in Geometric Group Theory within the curriculum of Mathematics Major at AUC and lead promising students into research in this field.