Hilbert's fifth problem asks whether every topological group which is locally Euclidean is in fact a Lie group. This problem was answered in the affirmative in the fifties following deeper investigations into the structure and representation theory of general locally compact topological groups. In this talk I will indicate how Hilbert's fifth is resolved and then we will consider a generalization of the problem which is still open: The Hilbert-Smith conjecture asks whether a locally compact group with a faithful, continuous action on a connected manifold is in fact a Lie group.

There is an intriguing, known reduction of the conjecture: It is sufficient to show that the locally compact group in question cannot be any of the (additive) groups of p-adic integers. In this talk I will explain why this is sufficient to prove the conjecture. Speaker(s): Andrew Odesky (UM)