Hilbert’s fifth problem, from his famous list of twenty-three problems in
mathematics from 1900, asks for a topological description of Lie groups,
without any direct reference to smooth structure. As with many of Hilbert’s
problems, this question can be formalised in a number of ways, but one com-
monly accepted formulation asks whether any locally Euclidean topological
group is necessarily a Lie group. This question was answered affirmatively by
Montgomery and Zippin [MoZi1952] and Gleason [Gl1952]; see Theorem
1.1.9. As a byproduct of the machinery developed to solve this problem, the
structure of locally compact groups was greatly clarified, leading in particu-
lar to the very useful Gleason-Yamabe theorem (Theorem 1.1.13) describing
such groups. This theorem (and related results) have since had a number
of applications, most strikingly in Gromov’s celebrated theorem [Gr1981]
on groups of polynomial growth (Theorem 1.3.1), and in the classification
of finite approximate groups (Theorem 1.2.12). These results in turn have
applications to the geometry of manifolds, and on related topics in geometric
group theory.
In the fall of 2011, I taught a graduate topics course covering these top-
ics, developed the machinery needed to solve Hilbert’s fifth problem, and
then used it to classify approximate groups and then finally to develop ap-
plications such as Gromov’s theorem. Along the way, one needs to develop
a number of standard mathematical tools, such as the Baker-Campbell-
Hausdorff formula relating the group law of a Lie group to the associated
Lie algebra, the Peter-Weyl theorem concerning the representation-theoretic
structure of a compact group, or the basic facts about ultrafilters and ultra-
products that underlie nonstandard analysis.
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