*Fearless Symmetry*"began as an expository article for professional mathematicians about the proof of Fermat's Last Theorem" but is, as its back-cover blurb claims, "written in a friendly style for a general audience". This seems like a contradiction and indeed

*Fearless Symmetry*is somewhat unbalanced.

The approach spans the largely unoccupied space between popular books and actual mathematics. Ash and Gross present key definitions and results, work through detailed illustrative examples, and attempt to provide intuition for key concepts. They largely omit the details of proofs, but stay focused on the mathematics, with pretty much no digressions to biography or history. What broader ruminations there are are attempts to give non-mathematicians a feel for concepts such as "representation" and "conjecture".

*Fearless Symmetry* proceeds in three parts of increasing difficulty.
Part one covers some "algebraic preliminaries". Chapters on
representations, groups, permutations, modular arithmetic, and complex
numbers assume almost no background and will be refresher material for
readers with any kind of training in formal mathematics. They provide
just enough foundation for what is to come, rather than attempting any
systematic presentation. Material on varieties and quadratic reciprocity
may be less familiar but still shouldn't pose any difficulty.

This is the foundation for part two, which is an introduction to Galois
Theory. This introduces the absolute Galois group of **Q**, linear
representations of Galois groups and their characters, and Frobenius
elements. As background to this, there's some basic material on matrices
and matrix multiplication and a rapid introduction to elliptic curves.

The difficulty level ramps up sharply in part three. This gives some detailed examples of how Galois representations and reciprocity laws can be used, looking at the roots of unity and various kinds of elliptic curves. It then presents some more algebraic number theory and uses that to get a new perspective on quadratic reciprocity, before very sketchily introducing étale cohomology and modular forms and attempting to place generalized reciprocity laws in a broader context. Finally the series of results leading up to Wiles' proof of Fermat's Last Theorem are outlined.

It's hard to know where *Fearless Symmetry* fits. Newcomers to
formal mathematics may enjoy part one but will probably find getting
through part two a challenge. Mathematicians, on the other hand, will
probably wish part three had been expanded at the expense of the rest.
*Fearless Symmetry* probably works best for those with some background
in mathematics. Falling in this category myself — I last looked at
Galois Theory twenty years ago — I found the foundational material
trivial but interesting in places, the core material on Galois Theory a
nice mix of the half-familiar and the novel, and the last half of part
three only barely comprehensible.

In any event, it's nice to see a book which tries to make actual
mathematics accessible to non-mathematicians, and which is prepared to
push readers out of their comfort zone. I recommend *Fearless Symmetry*
to anyone who enjoys mathematics and has some background in abstract
algebra, is curious about Fermat's Last Theorem, and doesn't mind
being confronted with some material that they probably aren't going
to understand.

July 2010

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- books about mathematics

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