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HORIZON-MSCA-2021-PF-01-01 MSCA Postdoctoral Fellowships 2021
Model theory, locally compact groups and solution of Hilbert's 5th problem
(Hilbert5th vs models)
Abstract: The main goal of the project is to apply advanced techniques from the model theory (a branch of mathematical logic) to the class of locally compact groups arising from the solution of Hilbert's 5th problem (so at the end, to the class of Lie groups), to answer the following question: how much geometry can model theory recognize? There does not exist a general (first-order) model-theoretic description of the locally compact groups, thus our first goal will be to develop such a description. Then, we will study how notions from these two corners of mathematics, i.e. model theory and locally compact groups, correspond to each other. For example, we will try to enrich the classification of locally compact and Lie groups by translating the dividing lines from the model-theoretic stability hierarchy. In the next stage, machinery from the so called geometric (neo)stability theory will be deployed in a tame class of locally compact groups, for example in the class of locally compact groups being projective limits of Lie groups and not having small subgroups (so in the groups from the solution of Hilbert's 5th problem). In this spirit, one could consider the definable homogeneous space coming from the Group Configuration Theorem, which is a part of the aforementioned machinery, and try to relate it to the unsolved Hilbert-Smith conjecture - this will be one of our milestones. In short, we aim to find connections between model-theoretic theorems of geometric nature and classical theorems on the Lie groups, so theorems which depend on the geometry of Lie groups. After understanding these connections, we want to transport techniques from the model theory into the locally compact and Lie groups and vice versa.
Host institution: Technische Universität Dresden
Host: Andreas Thom
Project period: Oct 2022 - Sep 2024
The project was financed by the European Union's Horizon 2020 research and innovation programme under the grant agreement no. 101063183
Invited guests
Alessandro Codenotti, June 12-16, 2023 at TUD
Kyle Gannon, January 13-23, 2024 at TUD
Tomasz Rzepecki, June 17-21, 2024 at TUD
Courses at TUD
Selected topics in model theory (module: Models and methods of pure mathematics - Geometrie)
Starting from stable theories, we went to NIP theories and discussed results of Chernikov and Simon on definably amenable NIP groups.
Dissemination of results
Articles
D.M. Hoffmann, On Weak Independence Theorem, RIMS Kôkyûroku, No. 2249, April 2023
J. Dobrowolski, D.M. Hoffmann, On rank not only in NSOP1 theories, Journal of Symbolic Logic (2024), doi: 10.1017/jsl.2024.9
A. Codenotti, D.M. Hoffmann, Ranks in Ellis semigroups and model theory (preprint)
D.M. Hoffmann, P. Kowalski, Ch.-M. Tran, J. Ye, Of model completeness and algebraic groups (preprint)
D.M. Hoffmann, T. Rzepecki Model theory of topological groups (in progress)
Talks and meetings
December 12- 14, 2022 Kyoto (Japan)
I attended RIMS Model Theory Workshop 2022, „Model theoretic aspects of the notion of independence and dimension” and gave there a talk „On Weak Independence Theorem” based on my previous results.
December 16-21, 2022 Będlewo (Poland)
It was a conference on the newest research in model theory. I met there several leading researchers in the field and consulted with them goals of my MSCA project. I got several suggestions, some very helpful.
January 8-14, 2023 Oberewolfach (Germany)
I participated in a workshop, „Model Theory: Combinatorics, Groups, Valued Fields and Neostability”, where – again – I was able to use the knowledge of the leading experts in the field to obtain feedback on my ideas in the MSCA.
January 28th, 2023 Bonn (Germany)
I was invited to give a talk at the Bonn-Münster-Düsseldorf GeSAMT (Gemeinsames Seminar Algebra und Modelltheorie) and I gave a talk „A bit of model theory of finite group actions”. During this meeting I started a research project with Alessandro Codenotti.
February 7th, 2023 Düsseldorf (Germany)
I was invited to give a talk at the research seminar in Düsseldorf. The title of the talk was „Interpreting Galois groups of many sorted structures“. After the talk I had a scientific discussion about the MSCA project goals.
May 17-20, 2023 Sirince (Turkey)
I participated in conference/workshop, Antalya Algebra Days 2023, where I gave a short talk „Shortly on Model Completeness of SL(2,R)”. The meeting was fruitful as I got new ideas for the MSCA project after scientific consultations.
June 5-6, 2023 Gdańsk (Poland)
I was invited to give a talk at the research seminar of the Mathematical Institute of the Polish Academy of Science in Sopot/Gdańsk. I delivered a talk „General on Model Completeness of SL(2,R)”.
August 15th, 2023 Shanghai (China, online)
I gave an online talk at the 2023 Fudan Conference on Mathematical Logic, the title of my talk was „Oscillations in space of types”.
August 28-30, 2023 Seoul (South Korea)
There was a conference, Model Theory Conference in Seoul. I spent my time there on discussions about model theory, groups and topological dynamics. I got good feedback from several key experts in the field.
September 15-23, 2023 Wrocław (Poland)
In that time there were two meetings in Wrocław: a workshop and a conference. I participated in both of them and delivered a warmly received talk „Oscillations in space of types”.
September 25-29, 2023 Münster (Germany)
I attended at the conference Model theory and groups (closely related to all goals of the MSCA project).
March 5-14, 2024 Wrocław (Poland)
I visited profesor Krzysztof Krupiński and his research group at the University of Wrocław. I gave two talks at the model theory seminar “Affine Keisler Dynamics” (part one & part two).
March 19-29, 2024 South Bend (USA)
I visited professor Anand Pillay and his research group at the University of Notre Dame in South Bend. We discussed several aspects related to topological dynamics and groups in model theory. I gave a talk at the model theory seminar, Affine Keisler Dynamics
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