Nonstandard Analysis and Hilbert's Fifth Problem
by
Isaac Goldbring
University of Illinois at Urbana Champaign

The most common version of Hilbert's Fifth Problem can be stated as "Is every locally euclidean topological group a Lie group?" More precisely, if G is a locally euclidean topological group, does there necessarily exist a compatible real analytic structure on G for which the group operations become real analytic? After affirmative answers to this question were given by Pontrjagin for the case that G is abelian and by von Neumann for the case that G is "linear", the full positive solution was achieved by Gleason, Montgomery, and Zippin in the early 1950s. In the early 1990s, Hirschfeld simplified much of their proof using nonstandard analysis, which is a way of approaching analysis by exploiting ideas from mathematical logic to provide a solid foundation for the use of infinitesimal elements. Recently, I extended Hirschfeld's methods to give a positive solution to the local version of Hilbert's Fifth Problem, which asks whether every locally euclidean local group is locally isomorphic to a Lie group. After introducing the fundamentals of nonstandard analysis, I will indicate how it is used in the solution of Hilbert's Fifth Problem as well as indicate why it was especially useful in the solution of the local version of Hilbert's Fifth Problem.

Paper reference: arXiv:0708.3871

Date received: January 24, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawj-08.