Hilbert5th vs models · Model theory, locally compact groups and solution of Hilbert's 5th problem
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2022-10-01 → 2024-09-30
- EU contribution
- €189,687
- Participants
- 2
- Scheme
- HORIZON-TMA-MSCA-PF-EF
Lines connect the coordinator with its partners.
Results in brief
Model theory, locally compact groups and solution of Hilbert's 5th problem
My research project investigates the intersection of model theory and the theory of locally compact groups. Hilbert's 5th problem (characterizing locally compact groups as Lie groups) is solved and thus deeper questions remain about their internal structure and the relationship to model-theoretic concepts. I aim to apply geometric stability theory to locally compact groups, developing a suitable first-order framework and identifying classes of groups exhibiting "tame" model-theoretic properties within stability hierarchy. This involves establishing a "dictionary" translating fundamental notions between model theory and Lie group theory. My project's expected impact is twofold: advancing the field of model theory by extending its scope to topological structures, and deepening our understanding of locally compact groups through a novel, model-theoretic lens. This interdisciplinary approach has the potential to yield significant advances, impacting related areas such as algebraic geometry and the study of definable groups.
Data: CORDIS, © European Union
Project objective
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.
Original text from CORDIS.
Participants
Links
- View on CORDIS
- DOI: 10.3030/101063183
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e50ac9809a&appId=PPGMS
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e515030126&appId=PPGMS
Data: CORDIS, © European Union