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.
