Logic Colloquium

Abstract. The most common version of Hilbert's Fifth Problem (H5) asks whether every locally euclidean topological group (i.e. a topological group whose identity has an open neighborhood homeomorphic to some R^n) can be given the structure of a real analytic manifold so that the group multiplication and inversion become real analytic maps; briefly, whether every locally euclidean topological group is a Lie group. A positive solution to the H5 was given by Gleason, Montgomery, and Zippin in the 1950s. Shortly after, Jacoby claimed to have proven that every locally euclidean "local group" was locally isomorphic to a Lie group. Roughly speaking, a local group is a hausdorff space which has a partial group-like operation defined on it which is continuous. However, Jacoby's proof was discovered to be flawed in the '90s, leaving some important theorems whose truth rests on this local version of the H5 on shaky foundation. By modifying techniques used by Hirschfeld in a proof of the H5 using nonstandard analysis, I was able to give a correct proof of the local H5. In this talk, I will try and sketch a part of the proof of the local H5. No background knowledge in Lie theory or nonstandard analysis will be necessary and all relevant terms will be defined.