xii Preface
This text is based on the lecture notes from that course, as well as from
some additional posts on my blog at terrytao.wordpress.com on further
topics related to Hilbert’s fifth problem. Part 1 of this text can thus serve
as the basis for a one-quarter or one-semester advanced graduate course,
depending on how much of the optional material one wishes to cover. The
material here assumes familiarity with basic graduate real analysis (such as
measure theory and point set topology), as covered for instance in my texts
[Ta2011], [Ta2010], and including topics such as the Riesz representation
theorem, the Arzel´ a-Ascoli theorem, Tychonoff’s theorem, and Urysohn’s
lemma. A basic understanding of linear algebra (including, for instance, the
spectral theorem for unitary matrices) is also assumed.
The core of the text is Part 1. The first part of this section of the
book is devoted to the theory surrounding Hilbert’s fifth problem, and in
particular in fleshing out the long road from locally compact groups to Lie
groups. First, the theory of Lie groups and Lie algebras is reviewed, and it is
shown that a Lie group structure can be built from a special type of metric
known as a Gleason metric, thanks to tools such as the Baker-Campbell-
Hausdorff formula. Some representation theory (and in particular, the Peter-
Weyl theorem) is introduced next, in order to classify compact groups. The
two tools are then combined to prove the fundamental Gleason-Yamabe
theorem, which among other things leads to a positive solution to Hilbert’s
fifth problem.
After this, the focus turns from the “soft analysis” of locally compact
groups to the “hard analysis” of approximate groups, with the useful tool of
ultraproducts serving as the key bridge between the two topics. By using this
bridge, one can start imposing approximate Lie structure on approximate
groups, which ultimately leads to a satisfactory classification of approximate
groups as well. Finally, Part 1 ends with applications of this classification
to geometric group theory and the geometry of manifolds, and in particular
in reproving Gromov’s theorem on groups of polynomial growth.
Part 2 contains a variety of additional material that is related to one
or more of the topics covered in Part 1, but which can be omitted for the
purposes of teaching a graduate course on the subject.
For reasons of space, we will not be able to define every single mathematical
term that we use in this book. If a term is italicised for reasons other than
emphasis or for definition, then it denotes a standard mathematical object,
result, or concept, which can be easily looked up in any number of references.
(In the blog version of the book, many of these terms were linked to their
Wikipedia pages, or other on-line reference pages.)
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