Periodic Reporting for period 1 - Hilbert5th vs models (Model theory, locally compact groups and solution of Hilbert's 5th problem)
Reporting period: 2022-10-01 to 2024-09-30
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.
During the project, I conducted research in pure mathematics. The initial phase focused on selecting an appropriate first-order structure to describe topological groups. After reviewing the literature, conducting independent analysis, and consulting with senior model theorists (Hrushovski, Newelski, Pillay, and Scanlon), I eliminated in my project approaches using continuous and semi-continuous logic due to issues concerning dimension theory of compact sets. Then, my secondment host (Tent) suggested adapting ideas from her research on the model theory of groups. This proved promising and, after modification, was successfully integrated into my project. The modified first-order structure - instead of using continuous multiplication of elements in a topological group, considers multiplication of small neighborhoods around each element. This natural approach clearly exploits local compactness to recover convergence arguments and led to several deeper results, including a method for encoding universal formulas from the (pure) group structure into my language. In a joint publication (with Kowalski, Tran, and Ye), I proved that many groups from classical Lie group theory are model complete, implying that only universal formulas matter, and therefore their theories are fully captured within my framework. Later, I proved a theorem showing that my framework can identify which locally compact groups are Lie groups. This result relies on the solution to Hilbert's 5th problem and the concept of the escape metric (Gleason's metric), where being a Lie group corresponds to realizing a type from a specific family of types I defined. Furthermore, my framework allows viewing the original locally compact group within its type space - this area is still under investigation with Rzepecki. Throughout the project, I also pursued advanced training in topological dynamics and its applications in model theory, including its connections to geometric group theory, to enhance my intuition and skills.
Main results (beyond the state of the art) in the project are:
1) Developing new first-order structure for topological groups within the model theory. Obtaining basic results for locally compact groups considered as structures in this new framework.
2) Theorem allowing to encode the entire theory of pure group structure in the new framework for groups which are model complete.
3) Theorem showing that many groups coming from geometry are model complete.
4) Theorem characterizing the class of Lie groups in the new framework.
5) Description of a locally compact group as a special subset of its space of types in the new framework.
Point (3) is the content of a preprint "Of model completeness and algebraic groups" (D.M. Hoffmann, P. Kowalski, Ch.-M. Tran, J. Ye), published on the arxiv server (a newer version will be uploaded soon and then submitted for publication in some leading scientific journal). Points (1), (2) and (4) are already written down in a form of a unfinished preprint. Point (5) exists in the form of notes (work in progress).
One of the main goals of the project was to establish a new framework to study topological groups via model-theoretic toolbox. This was accomplished in point (1). Studying this new framework and its limitations was the next goal and several interesting results were obtained (e.g. point (2) and (5)). Application of the new view on topological groups to the class of the well-known Lie groups was the key motivation behind point (4). Combining these results together we, see that the framework to study topological groups, which was developed in the project, is natural in the context of classical groups originating in geometry and can serve as a ground to deploy model-theoretic methods in the class of Lie groups. However, to fully accomplish this ambitious line of research, more time and bigger research time are needed.
Auxiliary results (beyond the state of the art) in the project are:
6) Developing tools in the geometric stability theory.
7) Studying topological dynamics in the context of model theory.
Point (6) is related to results in the article "On rank not only in NSOP1 theories" (J. Dobrowolski, D.M. Hoffmann) published in the Journal of Symbolic Logic. Small part of point (7) is the content of the preprint "Ranks in Ellis semigroups and model theory" (A. Codenotti, D.M. Hoffmann). Most of the output from point (7) is work-in-progress.
Results from point (6) extend the model-theoretic toolbox in the geometric stability theory by introducing new notion of a rank. The perspective application of the new rank is the description of generics for NSOP1 groups, which are only partially related to the main theme of the project. Anyway, obtaining new tools in model theory gave insights for the future applications of the framework developed in the project. The first stage of point (7) was concluded in a preprint, the next stages of point (7) exposed me to new techniques and to a new line of research, which extends the study of groups in model theory via methods from geometric group theory. Again, a strong research team and more funds are required to process in this direction of research - and therefore I applied for the ERC Start Grant focused on studying amenability of topological groups originating in model theory.
1) Developing new first-order structure for topological groups within the model theory. Obtaining basic results for locally compact groups considered as structures in this new framework.
2) Theorem allowing to encode the entire theory of pure group structure in the new framework for groups which are model complete.
3) Theorem showing that many groups coming from geometry are model complete.
4) Theorem characterizing the class of Lie groups in the new framework.
5) Description of a locally compact group as a special subset of its space of types in the new framework.
Point (3) is the content of a preprint "Of model completeness and algebraic groups" (D.M. Hoffmann, P. Kowalski, Ch.-M. Tran, J. Ye), published on the arxiv server (a newer version will be uploaded soon and then submitted for publication in some leading scientific journal). Points (1), (2) and (4) are already written down in a form of a unfinished preprint. Point (5) exists in the form of notes (work in progress).
One of the main goals of the project was to establish a new framework to study topological groups via model-theoretic toolbox. This was accomplished in point (1). Studying this new framework and its limitations was the next goal and several interesting results were obtained (e.g. point (2) and (5)). Application of the new view on topological groups to the class of the well-known Lie groups was the key motivation behind point (4). Combining these results together we, see that the framework to study topological groups, which was developed in the project, is natural in the context of classical groups originating in geometry and can serve as a ground to deploy model-theoretic methods in the class of Lie groups. However, to fully accomplish this ambitious line of research, more time and bigger research time are needed.
Auxiliary results (beyond the state of the art) in the project are:
6) Developing tools in the geometric stability theory.
7) Studying topological dynamics in the context of model theory.
Point (6) is related to results in the article "On rank not only in NSOP1 theories" (J. Dobrowolski, D.M. Hoffmann) published in the Journal of Symbolic Logic. Small part of point (7) is the content of the preprint "Ranks in Ellis semigroups and model theory" (A. Codenotti, D.M. Hoffmann). Most of the output from point (7) is work-in-progress.
Results from point (6) extend the model-theoretic toolbox in the geometric stability theory by introducing new notion of a rank. The perspective application of the new rank is the description of generics for NSOP1 groups, which are only partially related to the main theme of the project. Anyway, obtaining new tools in model theory gave insights for the future applications of the framework developed in the project. The first stage of point (7) was concluded in a preprint, the next stages of point (7) exposed me to new techniques and to a new line of research, which extends the study of groups in model theory via methods from geometric group theory. Again, a strong research team and more funds are required to process in this direction of research - and therefore I applied for the ERC Start Grant focused on studying amenability of topological groups originating in model theory.