Model theory, locally compact groups and solution of Hilbert's 5th problem

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.